Properties

Label 1617.4.a.j.1.2
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.37228 q^{2} -3.00000 q^{3} +3.37228 q^{4} +3.48913 q^{5} -10.1168 q^{6} -15.6060 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.37228 q^{2} -3.00000 q^{3} +3.37228 q^{4} +3.48913 q^{5} -10.1168 q^{6} -15.6060 q^{8} +9.00000 q^{9} +11.7663 q^{10} +11.0000 q^{11} -10.1168 q^{12} +15.0217 q^{13} -10.4674 q^{15} -79.6060 q^{16} -73.1684 q^{17} +30.3505 q^{18} +78.7011 q^{19} +11.7663 q^{20} +37.0951 q^{22} +112.000 q^{23} +46.8179 q^{24} -112.826 q^{25} +50.6576 q^{26} -27.0000 q^{27} +243.125 q^{29} -35.2989 q^{30} -278.717 q^{31} -143.606 q^{32} -33.0000 q^{33} -246.745 q^{34} +30.3505 q^{36} +102.380 q^{37} +265.402 q^{38} -45.0652 q^{39} -54.4512 q^{40} +241.255 q^{41} -280.016 q^{43} +37.0951 q^{44} +31.4021 q^{45} +377.696 q^{46} +169.870 q^{47} +238.818 q^{48} -380.481 q^{50} +219.505 q^{51} +50.6576 q^{52} -409.652 q^{53} -91.0516 q^{54} +38.3804 q^{55} -236.103 q^{57} +819.886 q^{58} -196.000 q^{59} -35.2989 q^{60} +701.359 q^{61} -939.913 q^{62} +152.568 q^{64} +52.4128 q^{65} -111.285 q^{66} +900.587 q^{67} -246.745 q^{68} -336.000 q^{69} +756.500 q^{71} -140.454 q^{72} +1019.81 q^{73} +345.255 q^{74} +338.478 q^{75} +265.402 q^{76} -151.973 q^{78} -327.549 q^{79} -277.755 q^{80} +81.0000 q^{81} +813.581 q^{82} +756.619 q^{83} -255.294 q^{85} -944.293 q^{86} -729.375 q^{87} -171.666 q^{88} -508.978 q^{89} +105.897 q^{90} +377.696 q^{92} +836.152 q^{93} +572.848 q^{94} +274.598 q^{95} +430.818 q^{96} -614.358 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 6 q^{3} + q^{4} - 16 q^{5} - 3 q^{6} + 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 6 q^{3} + q^{4} - 16 q^{5} - 3 q^{6} + 9 q^{8} + 18 q^{9} + 58 q^{10} + 22 q^{11} - 3 q^{12} + 76 q^{13} + 48 q^{15} - 119 q^{16} + 26 q^{17} + 9 q^{18} + 54 q^{19} + 58 q^{20} + 11 q^{22} + 224 q^{23} - 27 q^{24} + 142 q^{25} - 94 q^{26} - 54 q^{27} + 222 q^{29} - 174 q^{30} + 40 q^{31} - 247 q^{32} - 66 q^{33} - 482 q^{34} + 9 q^{36} - 48 q^{37} + 324 q^{38} - 228 q^{39} - 534 q^{40} + 494 q^{41} - 66 q^{43} + 11 q^{44} - 144 q^{45} + 112 q^{46} + 64 q^{47} + 357 q^{48} - 985 q^{50} - 78 q^{51} - 94 q^{52} - 84 q^{53} - 27 q^{54} - 176 q^{55} - 162 q^{57} + 870 q^{58} - 392 q^{59} - 174 q^{60} + 1104 q^{61} - 1696 q^{62} + 713 q^{64} - 1136 q^{65} - 33 q^{66} + 928 q^{67} - 482 q^{68} - 672 q^{69} + 456 q^{71} + 81 q^{72} + 592 q^{73} + 702 q^{74} - 426 q^{75} + 324 q^{76} + 282 q^{78} - 230 q^{79} + 490 q^{80} + 162 q^{81} + 214 q^{82} - 348 q^{83} - 2188 q^{85} - 1452 q^{86} - 666 q^{87} + 99 q^{88} - 972 q^{89} + 522 q^{90} + 112 q^{92} - 120 q^{93} + 824 q^{94} + 756 q^{95} + 741 q^{96} + 1184 q^{97} + 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.37228 1.19228 0.596141 0.802880i \(-0.296700\pi\)
0.596141 + 0.802880i \(0.296700\pi\)
\(3\) −3.00000 −0.577350
\(4\) 3.37228 0.421535
\(5\) 3.48913 0.312077 0.156038 0.987751i \(-0.450128\pi\)
0.156038 + 0.987751i \(0.450128\pi\)
\(6\) −10.1168 −0.688364
\(7\) 0 0
\(8\) −15.6060 −0.689693
\(9\) 9.00000 0.333333
\(10\) 11.7663 0.372083
\(11\) 11.0000 0.301511
\(12\) −10.1168 −0.243373
\(13\) 15.0217 0.320483 0.160242 0.987078i \(-0.448773\pi\)
0.160242 + 0.987078i \(0.448773\pi\)
\(14\) 0 0
\(15\) −10.4674 −0.180178
\(16\) −79.6060 −1.24384
\(17\) −73.1684 −1.04388 −0.521940 0.852982i \(-0.674791\pi\)
−0.521940 + 0.852982i \(0.674791\pi\)
\(18\) 30.3505 0.397427
\(19\) 78.7011 0.950277 0.475138 0.879911i \(-0.342398\pi\)
0.475138 + 0.879911i \(0.342398\pi\)
\(20\) 11.7663 0.131551
\(21\) 0 0
\(22\) 37.0951 0.359486
\(23\) 112.000 1.01537 0.507687 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 46.8179 0.398194
\(25\) −112.826 −0.902608
\(26\) 50.6576 0.382106
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 243.125 1.55680 0.778399 0.627769i \(-0.216032\pi\)
0.778399 + 0.627769i \(0.216032\pi\)
\(30\) −35.2989 −0.214822
\(31\) −278.717 −1.61481 −0.807405 0.589998i \(-0.799129\pi\)
−0.807405 + 0.589998i \(0.799129\pi\)
\(32\) −143.606 −0.793318
\(33\) −33.0000 −0.174078
\(34\) −246.745 −1.24460
\(35\) 0 0
\(36\) 30.3505 0.140512
\(37\) 102.380 0.454898 0.227449 0.973790i \(-0.426961\pi\)
0.227449 + 0.973790i \(0.426961\pi\)
\(38\) 265.402 1.13300
\(39\) −45.0652 −0.185031
\(40\) −54.4512 −0.215237
\(41\) 241.255 0.918970 0.459485 0.888186i \(-0.348034\pi\)
0.459485 + 0.888186i \(0.348034\pi\)
\(42\) 0 0
\(43\) −280.016 −0.993071 −0.496536 0.868016i \(-0.665395\pi\)
−0.496536 + 0.868016i \(0.665395\pi\)
\(44\) 37.0951 0.127098
\(45\) 31.4021 0.104026
\(46\) 377.696 1.21061
\(47\) 169.870 0.527192 0.263596 0.964633i \(-0.415091\pi\)
0.263596 + 0.964633i \(0.415091\pi\)
\(48\) 238.818 0.718133
\(49\) 0 0
\(50\) −380.481 −1.07616
\(51\) 219.505 0.602684
\(52\) 50.6576 0.135095
\(53\) −409.652 −1.06170 −0.530849 0.847466i \(-0.678127\pi\)
−0.530849 + 0.847466i \(0.678127\pi\)
\(54\) −91.0516 −0.229455
\(55\) 38.3804 0.0940947
\(56\) 0 0
\(57\) −236.103 −0.548643
\(58\) 819.886 1.85614
\(59\) −196.000 −0.432492 −0.216246 0.976339i \(-0.569381\pi\)
−0.216246 + 0.976339i \(0.569381\pi\)
\(60\) −35.2989 −0.0759512
\(61\) 701.359 1.47213 0.736064 0.676912i \(-0.236682\pi\)
0.736064 + 0.676912i \(0.236682\pi\)
\(62\) −939.913 −1.92531
\(63\) 0 0
\(64\) 152.568 0.297984
\(65\) 52.4128 0.100015
\(66\) −111.285 −0.207550
\(67\) 900.587 1.64215 0.821076 0.570819i \(-0.193374\pi\)
0.821076 + 0.570819i \(0.193374\pi\)
\(68\) −246.745 −0.440032
\(69\) −336.000 −0.586227
\(70\) 0 0
\(71\) 756.500 1.26451 0.632254 0.774762i \(-0.282130\pi\)
0.632254 + 0.774762i \(0.282130\pi\)
\(72\) −140.454 −0.229898
\(73\) 1019.81 1.63507 0.817536 0.575877i \(-0.195339\pi\)
0.817536 + 0.575877i \(0.195339\pi\)
\(74\) 345.255 0.542367
\(75\) 338.478 0.521121
\(76\) 265.402 0.400575
\(77\) 0 0
\(78\) −151.973 −0.220609
\(79\) −327.549 −0.466483 −0.233241 0.972419i \(-0.574933\pi\)
−0.233241 + 0.972419i \(0.574933\pi\)
\(80\) −277.755 −0.388175
\(81\) 81.0000 0.111111
\(82\) 813.581 1.09567
\(83\) 756.619 1.00060 0.500300 0.865852i \(-0.333223\pi\)
0.500300 + 0.865852i \(0.333223\pi\)
\(84\) 0 0
\(85\) −255.294 −0.325771
\(86\) −944.293 −1.18402
\(87\) −729.375 −0.898818
\(88\) −171.666 −0.207950
\(89\) −508.978 −0.606198 −0.303099 0.952959i \(-0.598021\pi\)
−0.303099 + 0.952959i \(0.598021\pi\)
\(90\) 105.897 0.124028
\(91\) 0 0
\(92\) 377.696 0.428016
\(93\) 836.152 0.932311
\(94\) 572.848 0.628561
\(95\) 274.598 0.296559
\(96\) 430.818 0.458023
\(97\) −614.358 −0.643079 −0.321539 0.946896i \(-0.604200\pi\)
−0.321539 + 0.946896i \(0.604200\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) −380.481 −0.380481
\(101\) 1015.92 1.00087 0.500434 0.865775i \(-0.333174\pi\)
0.500434 + 0.865775i \(0.333174\pi\)
\(102\) 740.234 0.718569
\(103\) −1102.16 −1.05436 −0.527181 0.849753i \(-0.676751\pi\)
−0.527181 + 0.849753i \(0.676751\pi\)
\(104\) −234.429 −0.221035
\(105\) 0 0
\(106\) −1381.46 −1.26584
\(107\) 1377.58 1.24463 0.622315 0.782767i \(-0.286192\pi\)
0.622315 + 0.782767i \(0.286192\pi\)
\(108\) −91.0516 −0.0811245
\(109\) 320.217 0.281388 0.140694 0.990053i \(-0.455067\pi\)
0.140694 + 0.990053i \(0.455067\pi\)
\(110\) 129.429 0.112187
\(111\) −307.141 −0.262636
\(112\) 0 0
\(113\) −1629.45 −1.35651 −0.678254 0.734828i \(-0.737263\pi\)
−0.678254 + 0.734828i \(0.737263\pi\)
\(114\) −796.206 −0.654136
\(115\) 390.782 0.316875
\(116\) 819.886 0.656245
\(117\) 135.196 0.106828
\(118\) −660.967 −0.515652
\(119\) 0 0
\(120\) 163.354 0.124267
\(121\) 121.000 0.0909091
\(122\) 2365.18 1.75519
\(123\) −723.766 −0.530568
\(124\) −939.913 −0.680699
\(125\) −829.805 −0.593760
\(126\) 0 0
\(127\) 2291.26 1.60091 0.800457 0.599390i \(-0.204590\pi\)
0.800457 + 0.599390i \(0.204590\pi\)
\(128\) 1663.35 1.14860
\(129\) 840.049 0.573350
\(130\) 176.751 0.119247
\(131\) 1147.41 0.765267 0.382633 0.923900i \(-0.375017\pi\)
0.382633 + 0.923900i \(0.375017\pi\)
\(132\) −111.285 −0.0733799
\(133\) 0 0
\(134\) 3037.03 1.95791
\(135\) −94.2064 −0.0600592
\(136\) 1141.86 0.719956
\(137\) 1268.60 0.791121 0.395561 0.918440i \(-0.370550\pi\)
0.395561 + 0.918440i \(0.370550\pi\)
\(138\) −1133.09 −0.698947
\(139\) 486.288 0.296737 0.148368 0.988932i \(-0.452598\pi\)
0.148368 + 0.988932i \(0.452598\pi\)
\(140\) 0 0
\(141\) −509.609 −0.304374
\(142\) 2551.13 1.50765
\(143\) 165.239 0.0966294
\(144\) −716.454 −0.414614
\(145\) 848.293 0.485841
\(146\) 3439.10 1.94947
\(147\) 0 0
\(148\) 345.255 0.191756
\(149\) 2354.11 1.29434 0.647169 0.762346i \(-0.275953\pi\)
0.647169 + 0.762346i \(0.275953\pi\)
\(150\) 1141.44 0.621323
\(151\) −570.070 −0.307229 −0.153615 0.988131i \(-0.549091\pi\)
−0.153615 + 0.988131i \(0.549091\pi\)
\(152\) −1228.21 −0.655399
\(153\) −658.516 −0.347960
\(154\) 0 0
\(155\) −972.479 −0.503945
\(156\) −151.973 −0.0779971
\(157\) 2072.67 1.05361 0.526807 0.849985i \(-0.323389\pi\)
0.526807 + 0.849985i \(0.323389\pi\)
\(158\) −1104.59 −0.556179
\(159\) 1228.96 0.612972
\(160\) −501.059 −0.247576
\(161\) 0 0
\(162\) 273.155 0.132476
\(163\) 2676.51 1.28614 0.643069 0.765808i \(-0.277661\pi\)
0.643069 + 0.765808i \(0.277661\pi\)
\(164\) 813.581 0.387378
\(165\) −115.141 −0.0543256
\(166\) 2551.53 1.19300
\(167\) 1188.12 0.550536 0.275268 0.961368i \(-0.411233\pi\)
0.275268 + 0.961368i \(0.411233\pi\)
\(168\) 0 0
\(169\) −1971.35 −0.897290
\(170\) −860.923 −0.388410
\(171\) 708.310 0.316759
\(172\) −944.293 −0.418615
\(173\) −807.147 −0.354718 −0.177359 0.984146i \(-0.556755\pi\)
−0.177359 + 0.984146i \(0.556755\pi\)
\(174\) −2459.66 −1.07164
\(175\) 0 0
\(176\) −875.666 −0.375033
\(177\) 588.000 0.249699
\(178\) −1716.42 −0.722758
\(179\) −1950.39 −0.814408 −0.407204 0.913337i \(-0.633496\pi\)
−0.407204 + 0.913337i \(0.633496\pi\)
\(180\) 105.897 0.0438505
\(181\) −1061.61 −0.435959 −0.217980 0.975953i \(-0.569947\pi\)
−0.217980 + 0.975953i \(0.569947\pi\)
\(182\) 0 0
\(183\) −2104.08 −0.849933
\(184\) −1747.87 −0.700297
\(185\) 357.218 0.141963
\(186\) 2819.74 1.11158
\(187\) −804.853 −0.314742
\(188\) 572.848 0.222230
\(189\) 0 0
\(190\) 926.021 0.353582
\(191\) 2136.41 0.809348 0.404674 0.914461i \(-0.367385\pi\)
0.404674 + 0.914461i \(0.367385\pi\)
\(192\) −457.704 −0.172041
\(193\) 3947.76 1.47236 0.736181 0.676784i \(-0.236627\pi\)
0.736181 + 0.676784i \(0.236627\pi\)
\(194\) −2071.79 −0.766731
\(195\) −157.238 −0.0577439
\(196\) 0 0
\(197\) 923.886 0.334133 0.167066 0.985946i \(-0.446571\pi\)
0.167066 + 0.985946i \(0.446571\pi\)
\(198\) 333.856 0.119829
\(199\) 476.152 0.169616 0.0848078 0.996397i \(-0.472972\pi\)
0.0848078 + 0.996397i \(0.472972\pi\)
\(200\) 1760.76 0.622522
\(201\) −2701.76 −0.948097
\(202\) 3425.96 1.19332
\(203\) 0 0
\(204\) 740.234 0.254053
\(205\) 841.770 0.286789
\(206\) −3716.80 −1.25710
\(207\) 1008.00 0.338458
\(208\) −1195.82 −0.398631
\(209\) 865.712 0.286519
\(210\) 0 0
\(211\) −4918.24 −1.60467 −0.802336 0.596872i \(-0.796410\pi\)
−0.802336 + 0.596872i \(0.796410\pi\)
\(212\) −1381.46 −0.447543
\(213\) −2269.50 −0.730064
\(214\) 4645.57 1.48395
\(215\) −977.012 −0.309915
\(216\) 421.361 0.132731
\(217\) 0 0
\(218\) 1079.86 0.335494
\(219\) −3059.44 −0.944010
\(220\) 129.429 0.0396642
\(221\) −1099.12 −0.334546
\(222\) −1035.77 −0.313136
\(223\) −2100.29 −0.630700 −0.315350 0.948975i \(-0.602122\pi\)
−0.315350 + 0.948975i \(0.602122\pi\)
\(224\) 0 0
\(225\) −1015.43 −0.300869
\(226\) −5494.95 −1.61734
\(227\) 2257.16 0.659970 0.329985 0.943986i \(-0.392956\pi\)
0.329985 + 0.943986i \(0.392956\pi\)
\(228\) −796.206 −0.231272
\(229\) 5311.07 1.53260 0.766301 0.642482i \(-0.222095\pi\)
0.766301 + 0.642482i \(0.222095\pi\)
\(230\) 1317.83 0.377804
\(231\) 0 0
\(232\) −3794.20 −1.07371
\(233\) 2466.27 0.693435 0.346718 0.937970i \(-0.387296\pi\)
0.346718 + 0.937970i \(0.387296\pi\)
\(234\) 455.918 0.127369
\(235\) 592.696 0.164524
\(236\) −660.967 −0.182311
\(237\) 982.646 0.269324
\(238\) 0 0
\(239\) 1429.40 0.386863 0.193432 0.981114i \(-0.438038\pi\)
0.193432 + 0.981114i \(0.438038\pi\)
\(240\) 833.266 0.224113
\(241\) 978.989 0.261669 0.130835 0.991404i \(-0.458234\pi\)
0.130835 + 0.991404i \(0.458234\pi\)
\(242\) 408.046 0.108389
\(243\) −243.000 −0.0641500
\(244\) 2365.18 0.620553
\(245\) 0 0
\(246\) −2440.74 −0.632586
\(247\) 1182.23 0.304548
\(248\) 4349.65 1.11372
\(249\) −2269.86 −0.577696
\(250\) −2798.33 −0.707929
\(251\) 6530.63 1.64227 0.821135 0.570734i \(-0.193341\pi\)
0.821135 + 0.570734i \(0.193341\pi\)
\(252\) 0 0
\(253\) 1232.00 0.306147
\(254\) 7726.76 1.90874
\(255\) 765.882 0.188084
\(256\) 4388.74 1.07147
\(257\) −8130.26 −1.97335 −0.986676 0.162696i \(-0.947981\pi\)
−0.986676 + 0.162696i \(0.947981\pi\)
\(258\) 2832.88 0.683595
\(259\) 0 0
\(260\) 176.751 0.0421600
\(261\) 2188.12 0.518933
\(262\) 3869.40 0.912414
\(263\) −4549.42 −1.06665 −0.533326 0.845910i \(-0.679058\pi\)
−0.533326 + 0.845910i \(0.679058\pi\)
\(264\) 514.997 0.120060
\(265\) −1429.33 −0.331332
\(266\) 0 0
\(267\) 1526.93 0.349988
\(268\) 3037.03 0.692225
\(269\) 29.1522 0.00660760 0.00330380 0.999995i \(-0.498948\pi\)
0.00330380 + 0.999995i \(0.498948\pi\)
\(270\) −317.690 −0.0716075
\(271\) −7711.22 −1.72850 −0.864250 0.503063i \(-0.832206\pi\)
−0.864250 + 0.503063i \(0.832206\pi\)
\(272\) 5824.64 1.29842
\(273\) 0 0
\(274\) 4278.07 0.943239
\(275\) −1241.09 −0.272147
\(276\) −1133.09 −0.247115
\(277\) 1127.52 0.244571 0.122286 0.992495i \(-0.460978\pi\)
0.122286 + 0.992495i \(0.460978\pi\)
\(278\) 1639.90 0.353794
\(279\) −2508.46 −0.538270
\(280\) 0 0
\(281\) −1872.47 −0.397517 −0.198758 0.980049i \(-0.563691\pi\)
−0.198758 + 0.980049i \(0.563691\pi\)
\(282\) −1718.54 −0.362900
\(283\) −2124.48 −0.446245 −0.223123 0.974790i \(-0.571625\pi\)
−0.223123 + 0.974790i \(0.571625\pi\)
\(284\) 2551.13 0.533034
\(285\) −823.794 −0.171219
\(286\) 557.233 0.115209
\(287\) 0 0
\(288\) −1292.45 −0.264439
\(289\) 440.621 0.0896846
\(290\) 2860.68 0.579259
\(291\) 1843.07 0.371282
\(292\) 3439.10 0.689241
\(293\) 3324.19 0.662802 0.331401 0.943490i \(-0.392479\pi\)
0.331401 + 0.943490i \(0.392479\pi\)
\(294\) 0 0
\(295\) −683.869 −0.134971
\(296\) −1597.75 −0.313740
\(297\) −297.000 −0.0580259
\(298\) 7938.73 1.54322
\(299\) 1682.44 0.325411
\(300\) 1141.44 0.219671
\(301\) 0 0
\(302\) −1922.44 −0.366304
\(303\) −3047.75 −0.577851
\(304\) −6265.07 −1.18200
\(305\) 2447.13 0.459417
\(306\) −2220.70 −0.414866
\(307\) 1698.94 0.315843 0.157921 0.987452i \(-0.449521\pi\)
0.157921 + 0.987452i \(0.449521\pi\)
\(308\) 0 0
\(309\) 3306.49 0.608736
\(310\) −3279.47 −0.600844
\(311\) −6928.83 −1.26334 −0.631668 0.775239i \(-0.717630\pi\)
−0.631668 + 0.775239i \(0.717630\pi\)
\(312\) 703.287 0.127615
\(313\) 3560.75 0.643020 0.321510 0.946906i \(-0.395810\pi\)
0.321510 + 0.946906i \(0.395810\pi\)
\(314\) 6989.64 1.25620
\(315\) 0 0
\(316\) −1104.59 −0.196639
\(317\) 332.750 0.0589561 0.0294780 0.999565i \(-0.490615\pi\)
0.0294780 + 0.999565i \(0.490615\pi\)
\(318\) 4144.39 0.730835
\(319\) 2674.37 0.469393
\(320\) 532.329 0.0929940
\(321\) −4132.73 −0.718587
\(322\) 0 0
\(323\) −5758.43 −0.991975
\(324\) 273.155 0.0468372
\(325\) −1694.84 −0.289271
\(326\) 9025.94 1.53344
\(327\) −960.652 −0.162459
\(328\) −3765.02 −0.633807
\(329\) 0 0
\(330\) −388.288 −0.0647714
\(331\) −541.445 −0.0899108 −0.0449554 0.998989i \(-0.514315\pi\)
−0.0449554 + 0.998989i \(0.514315\pi\)
\(332\) 2551.53 0.421788
\(333\) 921.423 0.151633
\(334\) 4006.67 0.656393
\(335\) 3142.26 0.512478
\(336\) 0 0
\(337\) 816.531 0.131986 0.0659930 0.997820i \(-0.478978\pi\)
0.0659930 + 0.997820i \(0.478978\pi\)
\(338\) −6647.94 −1.06982
\(339\) 4888.34 0.783180
\(340\) −860.923 −0.137324
\(341\) −3065.89 −0.486883
\(342\) 2388.62 0.377666
\(343\) 0 0
\(344\) 4369.92 0.684914
\(345\) −1172.35 −0.182948
\(346\) −2721.93 −0.422924
\(347\) 6260.53 0.968539 0.484269 0.874919i \(-0.339086\pi\)
0.484269 + 0.874919i \(0.339086\pi\)
\(348\) −2459.66 −0.378884
\(349\) 12768.5 1.95840 0.979198 0.202906i \(-0.0650386\pi\)
0.979198 + 0.202906i \(0.0650386\pi\)
\(350\) 0 0
\(351\) −405.587 −0.0616771
\(352\) −1579.67 −0.239194
\(353\) 2649.28 0.399453 0.199727 0.979852i \(-0.435995\pi\)
0.199727 + 0.979852i \(0.435995\pi\)
\(354\) 1982.90 0.297712
\(355\) 2639.52 0.394623
\(356\) −1716.42 −0.255534
\(357\) 0 0
\(358\) −6577.27 −0.971004
\(359\) −3203.91 −0.471020 −0.235510 0.971872i \(-0.575676\pi\)
−0.235510 + 0.971872i \(0.575676\pi\)
\(360\) −490.061 −0.0717457
\(361\) −665.143 −0.0969737
\(362\) −3580.04 −0.519786
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 3558.26 0.510268
\(366\) −7095.54 −1.01336
\(367\) 8429.40 1.19894 0.599470 0.800397i \(-0.295378\pi\)
0.599470 + 0.800397i \(0.295378\pi\)
\(368\) −8915.87 −1.26297
\(369\) 2171.30 0.306323
\(370\) 1204.64 0.169260
\(371\) 0 0
\(372\) 2819.74 0.393002
\(373\) −9388.53 −1.30327 −0.651635 0.758533i \(-0.725917\pi\)
−0.651635 + 0.758533i \(0.725917\pi\)
\(374\) −2714.19 −0.375261
\(375\) 2489.41 0.342807
\(376\) −2650.98 −0.363600
\(377\) 3652.16 0.498928
\(378\) 0 0
\(379\) −14264.5 −1.93329 −0.966647 0.256112i \(-0.917558\pi\)
−0.966647 + 0.256112i \(0.917558\pi\)
\(380\) 926.021 0.125010
\(381\) −6873.77 −0.924288
\(382\) 7204.58 0.964970
\(383\) −13462.2 −1.79605 −0.898026 0.439942i \(-0.854999\pi\)
−0.898026 + 0.439942i \(0.854999\pi\)
\(384\) −4990.05 −0.663144
\(385\) 0 0
\(386\) 13313.0 1.75547
\(387\) −2520.15 −0.331024
\(388\) −2071.79 −0.271080
\(389\) −941.881 −0.122764 −0.0613821 0.998114i \(-0.519551\pi\)
−0.0613821 + 0.998114i \(0.519551\pi\)
\(390\) −530.252 −0.0688470
\(391\) −8194.87 −1.05993
\(392\) 0 0
\(393\) −3442.24 −0.441827
\(394\) 3115.60 0.398380
\(395\) −1142.86 −0.145578
\(396\) 333.856 0.0423659
\(397\) 847.839 0.107183 0.0535917 0.998563i \(-0.482933\pi\)
0.0535917 + 0.998563i \(0.482933\pi\)
\(398\) 1605.72 0.202230
\(399\) 0 0
\(400\) 8981.62 1.12270
\(401\) 12203.6 1.51975 0.759875 0.650069i \(-0.225260\pi\)
0.759875 + 0.650069i \(0.225260\pi\)
\(402\) −9111.10 −1.13040
\(403\) −4186.82 −0.517520
\(404\) 3425.96 0.421901
\(405\) 282.619 0.0346752
\(406\) 0 0
\(407\) 1126.18 0.137157
\(408\) −3425.59 −0.415667
\(409\) −8759.53 −1.05900 −0.529500 0.848310i \(-0.677620\pi\)
−0.529500 + 0.848310i \(0.677620\pi\)
\(410\) 2838.69 0.341934
\(411\) −3805.79 −0.456754
\(412\) −3716.80 −0.444451
\(413\) 0 0
\(414\) 3399.26 0.403537
\(415\) 2639.94 0.312264
\(416\) −2157.21 −0.254245
\(417\) −1458.86 −0.171321
\(418\) 2919.42 0.341612
\(419\) 11188.4 1.30451 0.652256 0.757999i \(-0.273823\pi\)
0.652256 + 0.757999i \(0.273823\pi\)
\(420\) 0 0
\(421\) −14082.3 −1.63023 −0.815116 0.579298i \(-0.803327\pi\)
−0.815116 + 0.579298i \(0.803327\pi\)
\(422\) −16585.7 −1.91322
\(423\) 1528.83 0.175731
\(424\) 6393.02 0.732246
\(425\) 8255.30 0.942214
\(426\) −7653.39 −0.870441
\(427\) 0 0
\(428\) 4645.57 0.524655
\(429\) −495.718 −0.0557890
\(430\) −3294.76 −0.369505
\(431\) −5616.05 −0.627647 −0.313823 0.949481i \(-0.601610\pi\)
−0.313823 + 0.949481i \(0.601610\pi\)
\(432\) 2149.36 0.239378
\(433\) −7195.75 −0.798627 −0.399314 0.916814i \(-0.630752\pi\)
−0.399314 + 0.916814i \(0.630752\pi\)
\(434\) 0 0
\(435\) −2544.88 −0.280500
\(436\) 1079.86 0.118615
\(437\) 8814.52 0.964887
\(438\) −10317.3 −1.12553
\(439\) −101.959 −0.0110848 −0.00554240 0.999985i \(-0.501764\pi\)
−0.00554240 + 0.999985i \(0.501764\pi\)
\(440\) −598.963 −0.0648965
\(441\) 0 0
\(442\) −3706.53 −0.398873
\(443\) 4953.74 0.531285 0.265642 0.964072i \(-0.414416\pi\)
0.265642 + 0.964072i \(0.414416\pi\)
\(444\) −1035.77 −0.110710
\(445\) −1775.89 −0.189180
\(446\) −7082.78 −0.751972
\(447\) −7062.34 −0.747287
\(448\) 0 0
\(449\) −11602.0 −1.21945 −0.609723 0.792615i \(-0.708719\pi\)
−0.609723 + 0.792615i \(0.708719\pi\)
\(450\) −3424.33 −0.358721
\(451\) 2653.81 0.277080
\(452\) −5494.95 −0.571816
\(453\) 1710.21 0.177379
\(454\) 7611.79 0.786870
\(455\) 0 0
\(456\) 3684.62 0.378395
\(457\) −3530.68 −0.361397 −0.180698 0.983539i \(-0.557836\pi\)
−0.180698 + 0.983539i \(0.557836\pi\)
\(458\) 17910.4 1.82729
\(459\) 1975.55 0.200895
\(460\) 1317.83 0.133574
\(461\) −11566.3 −1.16854 −0.584271 0.811559i \(-0.698619\pi\)
−0.584271 + 0.811559i \(0.698619\pi\)
\(462\) 0 0
\(463\) 10888.5 1.09294 0.546470 0.837479i \(-0.315971\pi\)
0.546470 + 0.837479i \(0.315971\pi\)
\(464\) −19354.2 −1.93641
\(465\) 2917.44 0.290953
\(466\) 8316.94 0.826770
\(467\) −10688.0 −1.05906 −0.529529 0.848292i \(-0.677631\pi\)
−0.529529 + 0.848292i \(0.677631\pi\)
\(468\) 455.918 0.0450317
\(469\) 0 0
\(470\) 1998.74 0.196159
\(471\) −6218.02 −0.608304
\(472\) 3058.77 0.298287
\(473\) −3080.18 −0.299422
\(474\) 3313.76 0.321110
\(475\) −8879.53 −0.857728
\(476\) 0 0
\(477\) −3686.87 −0.353900
\(478\) 4820.35 0.461250
\(479\) −2341.90 −0.223391 −0.111696 0.993742i \(-0.535628\pi\)
−0.111696 + 0.993742i \(0.535628\pi\)
\(480\) 1503.18 0.142938
\(481\) 1537.93 0.145787
\(482\) 3301.43 0.311983
\(483\) 0 0
\(484\) 408.046 0.0383214
\(485\) −2143.57 −0.200690
\(486\) −819.464 −0.0764849
\(487\) 6748.91 0.627972 0.313986 0.949428i \(-0.398335\pi\)
0.313986 + 0.949428i \(0.398335\pi\)
\(488\) −10945.4 −1.01532
\(489\) −8029.53 −0.742552
\(490\) 0 0
\(491\) 7361.40 0.676609 0.338305 0.941037i \(-0.390147\pi\)
0.338305 + 0.941037i \(0.390147\pi\)
\(492\) −2440.74 −0.223653
\(493\) −17789.1 −1.62511
\(494\) 3986.80 0.363107
\(495\) 345.423 0.0313649
\(496\) 22187.6 2.00857
\(497\) 0 0
\(498\) −7654.60 −0.688777
\(499\) 10381.7 0.931359 0.465680 0.884953i \(-0.345810\pi\)
0.465680 + 0.884953i \(0.345810\pi\)
\(500\) −2798.33 −0.250291
\(501\) −3564.36 −0.317852
\(502\) 22023.1 1.95805
\(503\) −19149.0 −1.69744 −0.848721 0.528840i \(-0.822627\pi\)
−0.848721 + 0.528840i \(0.822627\pi\)
\(504\) 0 0
\(505\) 3544.67 0.312348
\(506\) 4154.65 0.365013
\(507\) 5914.04 0.518051
\(508\) 7726.76 0.674841
\(509\) −16073.2 −1.39967 −0.699836 0.714303i \(-0.746744\pi\)
−0.699836 + 0.714303i \(0.746744\pi\)
\(510\) 2582.77 0.224249
\(511\) 0 0
\(512\) 1493.27 0.128894
\(513\) −2124.93 −0.182881
\(514\) −27417.5 −2.35279
\(515\) −3845.58 −0.329042
\(516\) 2832.88 0.241687
\(517\) 1868.56 0.158954
\(518\) 0 0
\(519\) 2421.44 0.204797
\(520\) −817.952 −0.0689799
\(521\) 18955.3 1.59395 0.796975 0.604012i \(-0.206432\pi\)
0.796975 + 0.604012i \(0.206432\pi\)
\(522\) 7378.97 0.618714
\(523\) 4442.19 0.371402 0.185701 0.982606i \(-0.440544\pi\)
0.185701 + 0.982606i \(0.440544\pi\)
\(524\) 3869.40 0.322587
\(525\) 0 0
\(526\) −15341.9 −1.27175
\(527\) 20393.3 1.68567
\(528\) 2627.00 0.216525
\(529\) 377.000 0.0309855
\(530\) −4820.09 −0.395041
\(531\) −1764.00 −0.144164
\(532\) 0 0
\(533\) 3624.08 0.294515
\(534\) 5149.25 0.417285
\(535\) 4806.54 0.388420
\(536\) −14054.5 −1.13258
\(537\) 5851.17 0.470199
\(538\) 98.3096 0.00787812
\(539\) 0 0
\(540\) −317.690 −0.0253171
\(541\) 2180.90 0.173316 0.0866580 0.996238i \(-0.472381\pi\)
0.0866580 + 0.996238i \(0.472381\pi\)
\(542\) −26004.4 −2.06086
\(543\) 3184.82 0.251701
\(544\) 10507.4 0.828129
\(545\) 1117.28 0.0878146
\(546\) 0 0
\(547\) 8225.04 0.642920 0.321460 0.946923i \(-0.395826\pi\)
0.321460 + 0.946923i \(0.395826\pi\)
\(548\) 4278.07 0.333485
\(549\) 6312.23 0.490709
\(550\) −4185.29 −0.324475
\(551\) 19134.2 1.47939
\(552\) 5243.61 0.404316
\(553\) 0 0
\(554\) 3802.32 0.291598
\(555\) −1071.65 −0.0819625
\(556\) 1639.90 0.125085
\(557\) −25181.9 −1.91561 −0.957804 0.287423i \(-0.907201\pi\)
−0.957804 + 0.287423i \(0.907201\pi\)
\(558\) −8459.22 −0.641769
\(559\) −4206.33 −0.318263
\(560\) 0 0
\(561\) 2414.56 0.181716
\(562\) −6314.50 −0.473952
\(563\) 4504.50 0.337197 0.168599 0.985685i \(-0.446076\pi\)
0.168599 + 0.985685i \(0.446076\pi\)
\(564\) −1718.54 −0.128304
\(565\) −5685.34 −0.423335
\(566\) −7164.36 −0.532050
\(567\) 0 0
\(568\) −11805.9 −0.872122
\(569\) −13447.0 −0.990732 −0.495366 0.868684i \(-0.664966\pi\)
−0.495366 + 0.868684i \(0.664966\pi\)
\(570\) −2778.06 −0.204141
\(571\) −2605.52 −0.190959 −0.0954795 0.995431i \(-0.530438\pi\)
−0.0954795 + 0.995431i \(0.530438\pi\)
\(572\) 557.233 0.0407327
\(573\) −6409.24 −0.467277
\(574\) 0 0
\(575\) −12636.5 −0.916485
\(576\) 1373.11 0.0993281
\(577\) −6339.65 −0.457406 −0.228703 0.973496i \(-0.573448\pi\)
−0.228703 + 0.973496i \(0.573448\pi\)
\(578\) 1485.90 0.106929
\(579\) −11843.3 −0.850069
\(580\) 2860.68 0.204799
\(581\) 0 0
\(582\) 6215.37 0.442672
\(583\) −4506.17 −0.320114
\(584\) −15915.2 −1.12770
\(585\) 471.715 0.0333385
\(586\) 11210.1 0.790247
\(587\) 13370.6 0.940140 0.470070 0.882629i \(-0.344229\pi\)
0.470070 + 0.882629i \(0.344229\pi\)
\(588\) 0 0
\(589\) −21935.3 −1.53452
\(590\) −2306.20 −0.160923
\(591\) −2771.66 −0.192912
\(592\) −8150.09 −0.565822
\(593\) −14319.3 −0.991608 −0.495804 0.868434i \(-0.665127\pi\)
−0.495804 + 0.868434i \(0.665127\pi\)
\(594\) −1001.57 −0.0691832
\(595\) 0 0
\(596\) 7938.73 0.545609
\(597\) −1428.46 −0.0979276
\(598\) 5673.65 0.387981
\(599\) −5788.63 −0.394853 −0.197427 0.980318i \(-0.563258\pi\)
−0.197427 + 0.980318i \(0.563258\pi\)
\(600\) −5282.28 −0.359413
\(601\) −23968.1 −1.62675 −0.813375 0.581739i \(-0.802372\pi\)
−0.813375 + 0.581739i \(0.802372\pi\)
\(602\) 0 0
\(603\) 8105.28 0.547384
\(604\) −1922.44 −0.129508
\(605\) 422.184 0.0283706
\(606\) −10277.9 −0.688961
\(607\) 23526.6 1.57317 0.786585 0.617482i \(-0.211847\pi\)
0.786585 + 0.617482i \(0.211847\pi\)
\(608\) −11301.9 −0.753872
\(609\) 0 0
\(610\) 8252.40 0.547754
\(611\) 2551.74 0.168956
\(612\) −2220.70 −0.146677
\(613\) 1228.07 0.0809159 0.0404579 0.999181i \(-0.487118\pi\)
0.0404579 + 0.999181i \(0.487118\pi\)
\(614\) 5729.31 0.376573
\(615\) −2525.31 −0.165578
\(616\) 0 0
\(617\) −9844.90 −0.642368 −0.321184 0.947017i \(-0.604081\pi\)
−0.321184 + 0.947017i \(0.604081\pi\)
\(618\) 11150.4 0.725785
\(619\) 6551.68 0.425419 0.212709 0.977115i \(-0.431771\pi\)
0.212709 + 0.977115i \(0.431771\pi\)
\(620\) −3279.47 −0.212430
\(621\) −3024.00 −0.195409
\(622\) −23365.9 −1.50625
\(623\) 0 0
\(624\) 3587.46 0.230150
\(625\) 11208.0 0.717309
\(626\) 12007.8 0.766661
\(627\) −2597.14 −0.165422
\(628\) 6989.64 0.444135
\(629\) −7491.01 −0.474859
\(630\) 0 0
\(631\) −26440.5 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(632\) 5111.72 0.321730
\(633\) 14754.7 0.926458
\(634\) 1122.13 0.0702923
\(635\) 7994.48 0.499608
\(636\) 4144.39 0.258389
\(637\) 0 0
\(638\) 9018.74 0.559648
\(639\) 6808.50 0.421502
\(640\) 5803.64 0.358451
\(641\) −27927.2 −1.72084 −0.860421 0.509584i \(-0.829799\pi\)
−0.860421 + 0.509584i \(0.829799\pi\)
\(642\) −13936.7 −0.856758
\(643\) 16737.7 1.02655 0.513274 0.858225i \(-0.328432\pi\)
0.513274 + 0.858225i \(0.328432\pi\)
\(644\) 0 0
\(645\) 2931.03 0.178929
\(646\) −19419.1 −1.18271
\(647\) −7818.70 −0.475092 −0.237546 0.971376i \(-0.576343\pi\)
−0.237546 + 0.971376i \(0.576343\pi\)
\(648\) −1264.08 −0.0766325
\(649\) −2156.00 −0.130401
\(650\) −5715.49 −0.344892
\(651\) 0 0
\(652\) 9025.94 0.542152
\(653\) 19747.6 1.18344 0.591719 0.806144i \(-0.298450\pi\)
0.591719 + 0.806144i \(0.298450\pi\)
\(654\) −3239.59 −0.193697
\(655\) 4003.47 0.238822
\(656\) −19205.4 −1.14305
\(657\) 9178.33 0.545024
\(658\) 0 0
\(659\) 7867.72 0.465072 0.232536 0.972588i \(-0.425298\pi\)
0.232536 + 0.972588i \(0.425298\pi\)
\(660\) −388.288 −0.0229002
\(661\) −4227.41 −0.248755 −0.124378 0.992235i \(-0.539693\pi\)
−0.124378 + 0.992235i \(0.539693\pi\)
\(662\) −1825.90 −0.107199
\(663\) 3297.35 0.193150
\(664\) −11807.8 −0.690106
\(665\) 0 0
\(666\) 3107.30 0.180789
\(667\) 27230.0 1.58073
\(668\) 4006.67 0.232070
\(669\) 6300.88 0.364135
\(670\) 10596.6 0.611018
\(671\) 7714.94 0.443863
\(672\) 0 0
\(673\) 29397.6 1.68379 0.841897 0.539638i \(-0.181439\pi\)
0.841897 + 0.539638i \(0.181439\pi\)
\(674\) 2753.57 0.157364
\(675\) 3046.30 0.173707
\(676\) −6647.94 −0.378239
\(677\) −5737.14 −0.325696 −0.162848 0.986651i \(-0.552068\pi\)
−0.162848 + 0.986651i \(0.552068\pi\)
\(678\) 16484.8 0.933771
\(679\) 0 0
\(680\) 3984.11 0.224682
\(681\) −6771.49 −0.381034
\(682\) −10339.0 −0.580502
\(683\) 32097.6 1.79821 0.899107 0.437729i \(-0.144217\pi\)
0.899107 + 0.437729i \(0.144217\pi\)
\(684\) 2388.62 0.133525
\(685\) 4426.30 0.246891
\(686\) 0 0
\(687\) −15933.2 −0.884848
\(688\) 22291.0 1.23523
\(689\) −6153.69 −0.340257
\(690\) −3953.48 −0.218125
\(691\) 16456.2 0.905965 0.452983 0.891519i \(-0.350360\pi\)
0.452983 + 0.891519i \(0.350360\pi\)
\(692\) −2721.93 −0.149526
\(693\) 0 0
\(694\) 21112.3 1.15477
\(695\) 1696.72 0.0926047
\(696\) 11382.6 0.619909
\(697\) −17652.3 −0.959294
\(698\) 43058.9 2.33496
\(699\) −7398.80 −0.400355
\(700\) 0 0
\(701\) 27238.1 1.46758 0.733788 0.679379i \(-0.237751\pi\)
0.733788 + 0.679379i \(0.237751\pi\)
\(702\) −1367.75 −0.0735364
\(703\) 8057.44 0.432279
\(704\) 1678.25 0.0898457
\(705\) −1778.09 −0.0949882
\(706\) 8934.12 0.476261
\(707\) 0 0
\(708\) 1982.90 0.105257
\(709\) 28761.4 1.52349 0.761747 0.647875i \(-0.224342\pi\)
0.761747 + 0.647875i \(0.224342\pi\)
\(710\) 8901.21 0.470502
\(711\) −2947.94 −0.155494
\(712\) 7943.10 0.418090
\(713\) −31216.3 −1.63964
\(714\) 0 0
\(715\) 576.540 0.0301558
\(716\) −6577.27 −0.343302
\(717\) −4288.21 −0.223356
\(718\) −10804.5 −0.561588
\(719\) 27272.0 1.41456 0.707282 0.706931i \(-0.249921\pi\)
0.707282 + 0.706931i \(0.249921\pi\)
\(720\) −2499.80 −0.129392
\(721\) 0 0
\(722\) −2243.05 −0.115620
\(723\) −2936.97 −0.151075
\(724\) −3580.04 −0.183772
\(725\) −27430.8 −1.40518
\(726\) −1224.14 −0.0625785
\(727\) −3979.75 −0.203027 −0.101514 0.994834i \(-0.532369\pi\)
−0.101514 + 0.994834i \(0.532369\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 11999.5 0.608383
\(731\) 20488.3 1.03665
\(732\) −7095.54 −0.358277
\(733\) −9342.48 −0.470767 −0.235384 0.971903i \(-0.575635\pi\)
−0.235384 + 0.971903i \(0.575635\pi\)
\(734\) 28426.3 1.42947
\(735\) 0 0
\(736\) −16083.9 −0.805515
\(737\) 9906.45 0.495127
\(738\) 7322.23 0.365224
\(739\) −28928.0 −1.43997 −0.719983 0.693992i \(-0.755850\pi\)
−0.719983 + 0.693992i \(0.755850\pi\)
\(740\) 1204.64 0.0598425
\(741\) −3546.68 −0.175831
\(742\) 0 0
\(743\) 4857.04 0.239822 0.119911 0.992785i \(-0.461739\pi\)
0.119911 + 0.992785i \(0.461739\pi\)
\(744\) −13049.0 −0.643008
\(745\) 8213.80 0.403933
\(746\) −31660.8 −1.55386
\(747\) 6809.57 0.333533
\(748\) −2714.19 −0.132675
\(749\) 0 0
\(750\) 8395.00 0.408723
\(751\) 14355.4 0.697517 0.348759 0.937213i \(-0.386603\pi\)
0.348759 + 0.937213i \(0.386603\pi\)
\(752\) −13522.6 −0.655744
\(753\) −19591.9 −0.948165
\(754\) 12316.1 0.594863
\(755\) −1989.05 −0.0958792
\(756\) 0 0
\(757\) −17714.9 −0.850538 −0.425269 0.905067i \(-0.639821\pi\)
−0.425269 + 0.905067i \(0.639821\pi\)
\(758\) −48103.9 −2.30503
\(759\) −3696.00 −0.176754
\(760\) −4285.37 −0.204535
\(761\) 7945.82 0.378497 0.189248 0.981929i \(-0.439395\pi\)
0.189248 + 0.981929i \(0.439395\pi\)
\(762\) −23180.3 −1.10201
\(763\) 0 0
\(764\) 7204.58 0.341168
\(765\) −2297.64 −0.108590
\(766\) −45398.4 −2.14140
\(767\) −2944.26 −0.138606
\(768\) −13166.2 −0.618613
\(769\) −27308.1 −1.28057 −0.640284 0.768139i \(-0.721183\pi\)
−0.640284 + 0.768139i \(0.721183\pi\)
\(770\) 0 0
\(771\) 24390.8 1.13932
\(772\) 13313.0 0.620653
\(773\) 18872.6 0.878136 0.439068 0.898454i \(-0.355309\pi\)
0.439068 + 0.898454i \(0.355309\pi\)
\(774\) −8498.64 −0.394674
\(775\) 31446.6 1.45754
\(776\) 9587.65 0.443527
\(777\) 0 0
\(778\) −3176.29 −0.146369
\(779\) 18987.1 0.873276
\(780\) −530.252 −0.0243411
\(781\) 8321.50 0.381263
\(782\) −27635.4 −1.26373
\(783\) −6564.37 −0.299606
\(784\) 0 0
\(785\) 7231.82 0.328808
\(786\) −11608.2 −0.526782
\(787\) 14512.1 0.657307 0.328654 0.944450i \(-0.393405\pi\)
0.328654 + 0.944450i \(0.393405\pi\)
\(788\) 3115.60 0.140849
\(789\) 13648.3 0.615832
\(790\) −3854.04 −0.173570
\(791\) 0 0
\(792\) −1544.99 −0.0693167
\(793\) 10535.6 0.471792
\(794\) 2859.15 0.127793
\(795\) 4287.98 0.191294
\(796\) 1605.72 0.0714989
\(797\) −29108.9 −1.29371 −0.646856 0.762612i \(-0.723917\pi\)
−0.646856 + 0.762612i \(0.723917\pi\)
\(798\) 0 0
\(799\) −12429.1 −0.550325
\(800\) 16202.5 0.716056
\(801\) −4580.80 −0.202066
\(802\) 41154.0 1.81197
\(803\) 11218.0 0.492993
\(804\) −9111.10 −0.399656
\(805\) 0 0
\(806\) −14119.1 −0.617029
\(807\) −87.4567 −0.00381490
\(808\) −15854.4 −0.690291
\(809\) −3000.83 −0.130413 −0.0652063 0.997872i \(-0.520771\pi\)
−0.0652063 + 0.997872i \(0.520771\pi\)
\(810\) 953.071 0.0413426
\(811\) −6239.39 −0.270154 −0.135077 0.990835i \(-0.543128\pi\)
−0.135077 + 0.990835i \(0.543128\pi\)
\(812\) 0 0
\(813\) 23133.7 0.997950
\(814\) 3797.81 0.163530
\(815\) 9338.68 0.401374
\(816\) −17473.9 −0.749645
\(817\) −22037.6 −0.943693
\(818\) −29539.6 −1.26263
\(819\) 0 0
\(820\) 2838.69 0.120892
\(821\) 14922.4 0.634342 0.317171 0.948368i \(-0.397267\pi\)
0.317171 + 0.948368i \(0.397267\pi\)
\(822\) −12834.2 −0.544580
\(823\) −25737.8 −1.09011 −0.545057 0.838399i \(-0.683492\pi\)
−0.545057 + 0.838399i \(0.683492\pi\)
\(824\) 17200.3 0.727186
\(825\) 3723.26 0.157124
\(826\) 0 0
\(827\) 27043.4 1.13711 0.568555 0.822645i \(-0.307503\pi\)
0.568555 + 0.822645i \(0.307503\pi\)
\(828\) 3399.26 0.142672
\(829\) 9795.41 0.410384 0.205192 0.978722i \(-0.434218\pi\)
0.205192 + 0.978722i \(0.434218\pi\)
\(830\) 8902.62 0.372306
\(831\) −3382.56 −0.141203
\(832\) 2291.84 0.0954990
\(833\) 0 0
\(834\) −4919.70 −0.204263
\(835\) 4145.50 0.171809
\(836\) 2919.42 0.120778
\(837\) 7525.37 0.310770
\(838\) 37730.5 1.55535
\(839\) −28875.5 −1.18819 −0.594095 0.804395i \(-0.702490\pi\)
−0.594095 + 0.804395i \(0.702490\pi\)
\(840\) 0 0
\(841\) 34720.7 1.42362
\(842\) −47489.3 −1.94369
\(843\) 5617.41 0.229506
\(844\) −16585.7 −0.676426
\(845\) −6878.28 −0.280024
\(846\) 5155.63 0.209520
\(847\) 0 0
\(848\) 32610.7 1.32059
\(849\) 6373.45 0.257640
\(850\) 27839.2 1.12338
\(851\) 11466.6 0.461892
\(852\) −7653.39 −0.307747
\(853\) 47157.1 1.89288 0.946441 0.322878i \(-0.104650\pi\)
0.946441 + 0.322878i \(0.104650\pi\)
\(854\) 0 0
\(855\) 2471.38 0.0988531
\(856\) −21498.4 −0.858412
\(857\) −5021.31 −0.200145 −0.100073 0.994980i \(-0.531908\pi\)
−0.100073 + 0.994980i \(0.531908\pi\)
\(858\) −1671.70 −0.0665162
\(859\) 22921.1 0.910428 0.455214 0.890382i \(-0.349563\pi\)
0.455214 + 0.890382i \(0.349563\pi\)
\(860\) −3294.76 −0.130640
\(861\) 0 0
\(862\) −18938.9 −0.748332
\(863\) −19488.1 −0.768693 −0.384347 0.923189i \(-0.625573\pi\)
−0.384347 + 0.923189i \(0.625573\pi\)
\(864\) 3877.36 0.152674
\(865\) −2816.24 −0.110699
\(866\) −24266.1 −0.952188
\(867\) −1321.86 −0.0517794
\(868\) 0 0
\(869\) −3603.04 −0.140650
\(870\) −8582.05 −0.334435
\(871\) 13528.4 0.526282
\(872\) −4997.30 −0.194071
\(873\) −5529.22 −0.214360
\(874\) 29725.0 1.15042
\(875\) 0 0
\(876\) −10317.3 −0.397933
\(877\) −8455.67 −0.325573 −0.162787 0.986661i \(-0.552048\pi\)
−0.162787 + 0.986661i \(0.552048\pi\)
\(878\) −343.834 −0.0132162
\(879\) −9972.56 −0.382669
\(880\) −3055.31 −0.117039
\(881\) 11291.2 0.431794 0.215897 0.976416i \(-0.430732\pi\)
0.215897 + 0.976416i \(0.430732\pi\)
\(882\) 0 0
\(883\) 31818.1 1.21264 0.606322 0.795219i \(-0.292644\pi\)
0.606322 + 0.795219i \(0.292644\pi\)
\(884\) −3706.53 −0.141023
\(885\) 2051.61 0.0779254
\(886\) 16705.4 0.633441
\(887\) −17481.1 −0.661732 −0.330866 0.943678i \(-0.607341\pi\)
−0.330866 + 0.943678i \(0.607341\pi\)
\(888\) 4793.24 0.181138
\(889\) 0 0
\(890\) −5988.80 −0.225556
\(891\) 891.000 0.0335013
\(892\) −7082.78 −0.265862
\(893\) 13368.9 0.500978
\(894\) −23816.2 −0.890976
\(895\) −6805.16 −0.254158
\(896\) 0 0
\(897\) −5047.31 −0.187876
\(898\) −39125.1 −1.45392
\(899\) −67763.1 −2.51393
\(900\) −3424.33 −0.126827
\(901\) 29973.6 1.10829
\(902\) 8949.39 0.330357
\(903\) 0 0
\(904\) 25429.1 0.935574
\(905\) −3704.08 −0.136053
\(906\) 5767.31 0.211486
\(907\) 10607.4 0.388326 0.194163 0.980969i \(-0.437801\pi\)
0.194163 + 0.980969i \(0.437801\pi\)
\(908\) 7611.79 0.278201
\(909\) 9143.26 0.333623
\(910\) 0 0
\(911\) −41249.2 −1.50016 −0.750080 0.661347i \(-0.769985\pi\)
−0.750080 + 0.661347i \(0.769985\pi\)
\(912\) 18795.2 0.682425
\(913\) 8322.81 0.301692
\(914\) −11906.5 −0.430887
\(915\) −7341.38 −0.265244
\(916\) 17910.4 0.646045
\(917\) 0 0
\(918\) 6662.10 0.239523
\(919\) −13858.1 −0.497429 −0.248714 0.968577i \(-0.580008\pi\)
−0.248714 + 0.968577i \(0.580008\pi\)
\(920\) −6098.53 −0.218546
\(921\) −5096.83 −0.182352
\(922\) −39004.9 −1.39323
\(923\) 11363.9 0.405253
\(924\) 0 0
\(925\) −11551.2 −0.410595
\(926\) 36719.0 1.30309
\(927\) −9919.47 −0.351454
\(928\) −34914.2 −1.23504
\(929\) −20893.7 −0.737890 −0.368945 0.929451i \(-0.620281\pi\)
−0.368945 + 0.929451i \(0.620281\pi\)
\(930\) 9838.42 0.346897
\(931\) 0 0
\(932\) 8316.94 0.292307
\(933\) 20786.5 0.729388
\(934\) −36042.9 −1.26270
\(935\) −2808.23 −0.0982236
\(936\) −2109.86 −0.0736784
\(937\) −3203.52 −0.111691 −0.0558454 0.998439i \(-0.517785\pi\)
−0.0558454 + 0.998439i \(0.517785\pi\)
\(938\) 0 0
\(939\) −10682.2 −0.371248
\(940\) 1998.74 0.0693528
\(941\) −19951.6 −0.691182 −0.345591 0.938385i \(-0.612322\pi\)
−0.345591 + 0.938385i \(0.612322\pi\)
\(942\) −20968.9 −0.725270
\(943\) 27020.6 0.933099
\(944\) 15602.8 0.537952
\(945\) 0 0
\(946\) −10387.2 −0.356996
\(947\) −38216.7 −1.31138 −0.655689 0.755031i \(-0.727622\pi\)
−0.655689 + 0.755031i \(0.727622\pi\)
\(948\) 3313.76 0.113529
\(949\) 15319.4 0.524014
\(950\) −29944.3 −1.02265
\(951\) −998.249 −0.0340383
\(952\) 0 0
\(953\) −47661.4 −1.62004 −0.810022 0.586399i \(-0.800545\pi\)
−0.810022 + 0.586399i \(0.800545\pi\)
\(954\) −12433.2 −0.421948
\(955\) 7454.21 0.252579
\(956\) 4820.35 0.163077
\(957\) −8023.12 −0.271004
\(958\) −7897.56 −0.266345
\(959\) 0 0
\(960\) −1596.99 −0.0536901
\(961\) 47892.3 1.60761
\(962\) 5186.34 0.173819
\(963\) 12398.2 0.414876
\(964\) 3301.43 0.110303
\(965\) 13774.2 0.459490
\(966\) 0 0
\(967\) 18933.2 0.629628 0.314814 0.949153i \(-0.398058\pi\)
0.314814 + 0.949153i \(0.398058\pi\)
\(968\) −1888.32 −0.0626994
\(969\) 17275.3 0.572717
\(970\) −7228.73 −0.239279
\(971\) 40660.3 1.34382 0.671911 0.740632i \(-0.265474\pi\)
0.671911 + 0.740632i \(0.265474\pi\)
\(972\) −819.464 −0.0270415
\(973\) 0 0
\(974\) 22759.2 0.748720
\(975\) 5084.53 0.167011
\(976\) −55832.3 −1.83110
\(977\) −22502.8 −0.736876 −0.368438 0.929652i \(-0.620107\pi\)
−0.368438 + 0.929652i \(0.620107\pi\)
\(978\) −27077.8 −0.885331
\(979\) −5598.76 −0.182775
\(980\) 0 0
\(981\) 2881.96 0.0937959
\(982\) 24824.7 0.806709
\(983\) 4435.20 0.143907 0.0719536 0.997408i \(-0.477077\pi\)
0.0719536 + 0.997408i \(0.477077\pi\)
\(984\) 11295.1 0.365929
\(985\) 3223.55 0.104275
\(986\) −59989.8 −1.93759
\(987\) 0 0
\(988\) 3986.80 0.128378
\(989\) −31361.8 −1.00834
\(990\) 1164.86 0.0373958
\(991\) 7362.76 0.236010 0.118005 0.993013i \(-0.462350\pi\)
0.118005 + 0.993013i \(0.462350\pi\)
\(992\) 40025.5 1.28106
\(993\) 1624.33 0.0519101
\(994\) 0 0
\(995\) 1661.35 0.0529331
\(996\) −7654.60 −0.243519
\(997\) 53480.1 1.69883 0.849413 0.527728i \(-0.176956\pi\)
0.849413 + 0.527728i \(0.176956\pi\)
\(998\) 35010.0 1.11044
\(999\) −2764.27 −0.0875452
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 1617.4.a.j.1.2 2
7.6 odd 2 33.4.a.d.1.2 2
21.20 even 2 99.4.a.e.1.1 2
28.27 even 2 528.4.a.o.1.1 2
35.13 even 4 825.4.c.i.199.1 4
35.27 even 4 825.4.c.i.199.4 4
35.34 odd 2 825.4.a.k.1.1 2
56.13 odd 2 2112.4.a.ba.1.2 2
56.27 even 2 2112.4.a.bh.1.2 2
77.76 even 2 363.4.a.j.1.1 2
84.83 odd 2 1584.4.a.x.1.2 2
105.104 even 2 2475.4.a.o.1.2 2
231.230 odd 2 1089.4.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 7.6 odd 2
99.4.a.e.1.1 2 21.20 even 2
363.4.a.j.1.1 2 77.76 even 2
528.4.a.o.1.1 2 28.27 even 2
825.4.a.k.1.1 2 35.34 odd 2
825.4.c.i.199.1 4 35.13 even 4
825.4.c.i.199.4 4 35.27 even 4
1089.4.a.t.1.2 2 231.230 odd 2
1584.4.a.x.1.2 2 84.83 odd 2
1617.4.a.j.1.2 2 1.1 even 1 trivial
2112.4.a.ba.1.2 2 56.13 odd 2
2112.4.a.bh.1.2 2 56.27 even 2
2475.4.a.o.1.2 2 105.104 even 2