Properties

Label 33.4.a.d.1.2
Level $33$
Weight $4$
Character 33.1
Self dual yes
Analytic conductor $1.947$
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] = [33,4,Mod(1,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, 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("33.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(1.94706303019\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} +10.1168 q^{6} -4.74456 q^{7} -15.6060 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} +10.1168 q^{6} -4.74456 q^{7} -15.6060 q^{8} +9.00000 q^{9} -11.7663 q^{10} +11.0000 q^{11} +10.1168 q^{12} -15.0217 q^{13} -16.0000 q^{14} -10.4674 q^{15} -79.6060 q^{16} +73.1684 q^{17} +30.3505 q^{18} -78.7011 q^{19} -11.7663 q^{20} -14.2337 q^{21} +37.0951 q^{22} +112.000 q^{23} -46.8179 q^{24} -112.826 q^{25} -50.6576 q^{26} +27.0000 q^{27} -16.0000 q^{28} +243.125 q^{29} -35.2989 q^{30} +278.717 q^{31} -143.606 q^{32} +33.0000 q^{33} +246.745 q^{34} +16.5544 q^{35} +30.3505 q^{36} +102.380 q^{37} -265.402 q^{38} -45.0652 q^{39} +54.4512 q^{40} -241.255 q^{41} -48.0000 q^{42} -280.016 q^{43} +37.0951 q^{44} -31.4021 q^{45} +377.696 q^{46} -169.870 q^{47} -238.818 q^{48} -320.489 q^{49} -380.481 q^{50} +219.505 q^{51} -50.6576 q^{52} -409.652 q^{53} +91.0516 q^{54} -38.3804 q^{55} +74.0435 q^{56} -236.103 q^{57} +819.886 q^{58} +196.000 q^{59} -35.2989 q^{60} -701.359 q^{61} +939.913 q^{62} -42.7011 q^{63} +152.568 q^{64} +52.4128 q^{65} +111.285 q^{66} +900.587 q^{67} +246.745 q^{68} +336.000 q^{69} +55.8260 q^{70} +756.500 q^{71} -140.454 q^{72} -1019.81 q^{73} +345.255 q^{74} -338.478 q^{75} -265.402 q^{76} -52.1902 q^{77} -151.973 q^{78} -327.549 q^{79} +277.755 q^{80} +81.0000 q^{81} -813.581 q^{82} -756.619 q^{83} -48.0000 q^{84} -255.294 q^{85} -944.293 q^{86} +729.375 q^{87} -171.666 q^{88} +508.978 q^{89} -105.897 q^{90} +71.2716 q^{91} +377.696 q^{92} +836.152 q^{93} -572.848 q^{94} +274.598 q^{95} -430.818 q^{96} +614.358 q^{97} -1080.78 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9} - 58 q^{10} + 22 q^{11} + 3 q^{12} - 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} - 26 q^{17} + 9 q^{18} - 54 q^{19} - 58 q^{20} + 6 q^{21} + 11 q^{22} + 224 q^{23} + 27 q^{24} + 142 q^{25} + 94 q^{26} + 54 q^{27} - 32 q^{28} + 222 q^{29} - 174 q^{30} - 40 q^{31} - 247 q^{32} + 66 q^{33} + 482 q^{34} + 148 q^{35} + 9 q^{36} - 48 q^{37} - 324 q^{38} - 228 q^{39} + 534 q^{40} - 494 q^{41} - 96 q^{42} - 66 q^{43} + 11 q^{44} + 144 q^{45} + 112 q^{46} - 64 q^{47} - 357 q^{48} - 618 q^{49} - 985 q^{50} - 78 q^{51} + 94 q^{52} - 84 q^{53} + 27 q^{54} + 176 q^{55} + 240 q^{56} - 162 q^{57} + 870 q^{58} + 392 q^{59} - 174 q^{60} - 1104 q^{61} + 1696 q^{62} + 18 q^{63} + 713 q^{64} - 1136 q^{65} + 33 q^{66} + 928 q^{67} + 482 q^{68} + 672 q^{69} - 256 q^{70} + 456 q^{71} + 81 q^{72} - 592 q^{73} + 702 q^{74} + 426 q^{75} - 324 q^{76} + 22 q^{77} + 282 q^{78} - 230 q^{79} - 490 q^{80} + 162 q^{81} - 214 q^{82} + 348 q^{83} - 96 q^{84} - 2188 q^{85} - 1452 q^{86} + 666 q^{87} + 99 q^{88} + 972 q^{89} - 522 q^{90} - 340 q^{91} + 112 q^{92} - 120 q^{93} - 824 q^{94} + 756 q^{95} - 741 q^{96} - 1184 q^{97} - 375 q^{98} + 198 q^{99}+O(q^{100}) \) 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 \(-0.549872\pi\)
−0.156038 + 0.987751i \(0.549872\pi\)
\(6\) 10.1168 0.688364
\(7\) −4.74456 −0.256182 −0.128091 0.991762i \(-0.540885\pi\)
−0.128091 + 0.991762i \(0.540885\pi\)
\(8\) −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 \(-0.551227\pi\)
−0.160242 + 0.987078i \(0.551227\pi\)
\(14\) −16.0000 −0.305441
\(15\) −10.4674 −0.180178
\(16\) −79.6060 −1.24384
\(17\) 73.1684 1.04388 0.521940 0.852982i \(-0.325209\pi\)
0.521940 + 0.852982i \(0.325209\pi\)
\(18\) 30.3505 0.397427
\(19\) −78.7011 −0.950277 −0.475138 0.879911i \(-0.657602\pi\)
−0.475138 + 0.879911i \(0.657602\pi\)
\(20\) −11.7663 −0.131551
\(21\) −14.2337 −0.147907
\(22\) 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
\(28\) −16.0000 −0.107990
\(29\) 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 \(-0.200871\pi\)
0.807405 + 0.589998i \(0.200871\pi\)
\(32\) −143.606 −0.793318
\(33\) 33.0000 0.174078
\(34\) 246.745 1.24460
\(35\) 16.5544 0.0799486
\(36\) 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 \(-0.651966\pi\)
−0.459485 + 0.888186i \(0.651966\pi\)
\(42\) −48.0000 −0.176347
\(43\) −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 \(-0.584909\pi\)
−0.263596 + 0.964633i \(0.584909\pi\)
\(48\) −238.818 −0.718133
\(49\) −320.489 −0.934371
\(50\) −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
\(56\) 74.0435 0.176687
\(57\) −236.103 −0.548643
\(58\) 819.886 1.85614
\(59\) 196.000 0.432492 0.216246 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) −35.2989 −0.0759512
\(61\) −701.359 −1.47213 −0.736064 0.676912i \(-0.763318\pi\)
−0.736064 + 0.676912i \(0.763318\pi\)
\(62\) 939.913 1.92531
\(63\) −42.7011 −0.0853941
\(64\) 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
\(70\) 55.8260 0.0953212
\(71\) 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 \(-0.804661\pi\)
−0.817536 + 0.575877i \(0.804661\pi\)
\(74\) 345.255 0.542367
\(75\) −338.478 −0.521121
\(76\) −265.402 −0.400575
\(77\) −52.1902 −0.0772419
\(78\) −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 \(-0.666777\pi\)
−0.500300 + 0.865852i \(0.666777\pi\)
\(84\) −48.0000 −0.0623480
\(85\) −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 \(-0.401979\pi\)
0.303099 + 0.952959i \(0.401979\pi\)
\(90\) −105.897 −0.124028
\(91\) 71.2716 0.0821022
\(92\) 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 \(-0.395800\pi\)
0.321539 + 0.946896i \(0.395800\pi\)
\(98\) −1080.78 −1.11403
\(99\) 99.0000 0.100504
\(100\) −380.481 −0.380481
\(101\) −1015.92 −1.00087 −0.500434 0.865775i \(-0.666826\pi\)
−0.500434 + 0.865775i \(0.666826\pi\)
\(102\) 740.234 0.718569
\(103\) 1102.16 1.05436 0.527181 0.849753i \(-0.323249\pi\)
0.527181 + 0.849753i \(0.323249\pi\)
\(104\) 234.429 0.221035
\(105\) 49.6631 0.0461583
\(106\) −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
\(112\) 377.696 0.318651
\(113\) −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
\(119\) −347.152 −0.267423
\(120\) 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
\(126\) −144.000 −0.101814
\(127\) 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 \(-0.624983\pi\)
−0.382633 + 0.923900i \(0.624983\pi\)
\(132\) 111.285 0.0733799
\(133\) 373.402 0.243444
\(134\) 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 \(-0.547402\pi\)
−0.148368 + 0.988932i \(0.547402\pi\)
\(140\) 55.8260 0.0337011
\(141\) −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
\(147\) −961.467 −0.539459
\(148\) 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
\(154\) −176.000 −0.0920941
\(155\) −972.479 −0.503945
\(156\) −151.973 −0.0779971
\(157\) −2072.67 −1.05361 −0.526807 0.849985i \(-0.676611\pi\)
−0.526807 + 0.849985i \(0.676611\pi\)
\(158\) −1104.59 −0.556179
\(159\) −1228.96 −0.612972
\(160\) 501.059 0.247576
\(161\) −531.391 −0.260121
\(162\) 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 \(-0.588767\pi\)
−0.275268 + 0.961368i \(0.588767\pi\)
\(168\) 222.130 0.102010
\(169\) −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 \(-0.443245\pi\)
0.177359 + 0.984146i \(0.443245\pi\)
\(174\) 2459.66 1.07164
\(175\) 535.310 0.231232
\(176\) −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 \(-0.430053\pi\)
0.217980 + 0.975953i \(0.430053\pi\)
\(182\) 240.348 0.0978889
\(183\) −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
\(189\) −128.103 −0.0493023
\(190\) 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
\(196\) −1080.78 −0.393870
\(197\) 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 \(-0.527028\pi\)
−0.0848078 + 0.996397i \(0.527028\pi\)
\(200\) 1760.76 0.622522
\(201\) 2701.76 0.948097
\(202\) −3425.96 −1.19332
\(203\) −1153.52 −0.398824
\(204\) 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
\(210\) 167.478 0.0550337
\(211\) −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
\(217\) −1322.39 −0.413686
\(218\) 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 \(-0.397878\pi\)
0.315350 + 0.948975i \(0.397878\pi\)
\(224\) 681.348 0.203234
\(225\) −1015.43 −0.300869
\(226\) −5494.95 −1.61734
\(227\) −2257.16 −0.659970 −0.329985 0.943986i \(-0.607044\pi\)
−0.329985 + 0.943986i \(0.607044\pi\)
\(228\) −796.206 −0.231272
\(229\) −5311.07 −1.53260 −0.766301 0.642482i \(-0.777905\pi\)
−0.766301 + 0.642482i \(0.777905\pi\)
\(230\) −1317.83 −0.377804
\(231\) −156.571 −0.0445956
\(232\) −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
\(238\) −1170.70 −0.318844
\(239\) 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 \(-0.541766\pi\)
−0.130835 + 0.991404i \(0.541766\pi\)
\(242\) 408.046 0.108389
\(243\) 243.000 0.0641500
\(244\) −2365.18 −0.620553
\(245\) 1118.23 0.291595
\(246\) −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 \(-0.806659\pi\)
−0.821135 + 0.570734i \(0.806659\pi\)
\(252\) −144.000 −0.0359966
\(253\) 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 \(-0.0520188\pi\)
0.986676 + 0.162696i \(0.0520188\pi\)
\(258\) −2832.88 −0.683595
\(259\) −485.750 −0.116537
\(260\) 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
\(266\) 1259.22 0.290254
\(267\) 1526.93 0.349988
\(268\) 3037.03 0.692225
\(269\) −29.1522 −0.00660760 −0.00330380 0.999995i \(-0.501052\pi\)
−0.00330380 + 0.999995i \(0.501052\pi\)
\(270\) −317.690 −0.0716075
\(271\) 7711.22 1.72850 0.864250 0.503063i \(-0.167794\pi\)
0.864250 + 0.503063i \(0.167794\pi\)
\(272\) −5824.64 −1.29842
\(273\) 213.815 0.0474017
\(274\) 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
\(280\) −258.347 −0.0551400
\(281\) −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 \(-0.428375\pi\)
0.223123 + 0.974790i \(0.428375\pi\)
\(284\) 2551.13 0.533034
\(285\) 823.794 0.171219
\(286\) −557.233 −0.115209
\(287\) 1144.65 0.235424
\(288\) −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 \(-0.607521\pi\)
−0.331401 + 0.943490i \(0.607521\pi\)
\(294\) −3242.34 −0.643187
\(295\) −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
\(301\) 1328.55 0.254407
\(302\) −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 \(-0.550479\pi\)
−0.157921 + 0.987452i \(0.550479\pi\)
\(308\) −176.000 −0.0325602
\(309\) 3306.49 0.608736
\(310\) −3279.47 −0.600844
\(311\) 6928.83 1.26334 0.631668 0.775239i \(-0.282370\pi\)
0.631668 + 0.775239i \(0.282370\pi\)
\(312\) 703.287 0.127615
\(313\) −3560.75 −0.643020 −0.321510 0.946906i \(-0.604190\pi\)
−0.321510 + 0.946906i \(0.604190\pi\)
\(314\) −6989.64 −1.25620
\(315\) 148.989 0.0266495
\(316\) −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
\(322\) −1792.00 −0.310137
\(323\) −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
\(329\) 805.957 0.135057
\(330\) −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
\(336\) 1133.09 0.183973
\(337\) 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
\(343\) 3147.97 0.495552
\(344\) 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 \(-0.934961\pi\)
−0.979198 + 0.202906i \(0.934961\pi\)
\(350\) 1805.22 0.275694
\(351\) −405.587 −0.0616771
\(352\) −1579.67 −0.239194
\(353\) −2649.28 −0.399453 −0.199727 0.979852i \(-0.564005\pi\)
−0.199727 + 0.979852i \(0.564005\pi\)
\(354\) 1982.90 0.297712
\(355\) −2639.52 −0.394623
\(356\) 1716.42 0.255534
\(357\) −1041.46 −0.154397
\(358\) −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
\(364\) 240.348 0.0346089
\(365\) 3558.26 0.510268
\(366\) −7095.54 −1.01336
\(367\) −8429.40 −1.19894 −0.599470 0.800397i \(-0.704622\pi\)
−0.599470 + 0.800397i \(0.704622\pi\)
\(368\) −8915.87 −1.26297
\(369\) −2171.30 −0.306323
\(370\) −1204.64 −0.169260
\(371\) 1943.62 0.271988
\(372\) 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
\(378\) −432.000 −0.0587822
\(379\) −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 \(-0.145001\pi\)
0.898026 + 0.439942i \(0.145001\pi\)
\(384\) 4990.05 0.663144
\(385\) 182.098 0.0241054
\(386\) 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
\(392\) 5001.54 0.644429
\(393\) −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 \(-0.517067\pi\)
−0.0535917 + 0.998563i \(0.517067\pi\)
\(398\) −1605.72 −0.202230
\(399\) 1120.21 0.140553
\(400\) 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
\(406\) −3890.00 −0.475511
\(407\) 1126.18 0.137157
\(408\) −3425.59 −0.415667
\(409\) 8759.53 1.05900 0.529500 0.848310i \(-0.322380\pi\)
0.529500 + 0.848310i \(0.322380\pi\)
\(410\) 2838.69 0.341934
\(411\) 3805.79 0.456754
\(412\) 3716.80 0.444451
\(413\) −929.934 −0.110797
\(414\) 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 \(-0.726177\pi\)
−0.652256 + 0.757999i \(0.726177\pi\)
\(420\) 167.478 0.0194574
\(421\) −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
\(427\) 3327.64 0.377133
\(428\) 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 \(-0.369248\pi\)
0.399314 + 0.916814i \(0.369248\pi\)
\(434\) −4459.48 −0.493230
\(435\) −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 \(-0.498236\pi\)
0.00554240 + 0.999985i \(0.498236\pi\)
\(440\) 598.963 0.0648965
\(441\) −2884.40 −0.311457
\(442\) −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
\(448\) −723.869 −0.0763383
\(449\) −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
\(455\) −248.676 −0.0256222
\(456\) 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 \(-0.301381\pi\)
0.584271 + 0.811559i \(0.301381\pi\)
\(462\) −528.000 −0.0531705
\(463\) 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 \(-0.322369\pi\)
0.529529 + 0.848292i \(0.322369\pi\)
\(468\) −455.918 −0.0450317
\(469\) −4272.89 −0.420690
\(470\) 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
\(476\) −1170.70 −0.112728
\(477\) −3686.87 −0.353900
\(478\) 4820.35 0.461250
\(479\) 2341.90 0.223391 0.111696 0.993742i \(-0.464372\pi\)
0.111696 + 0.993742i \(0.464372\pi\)
\(480\) 1503.18 0.142938
\(481\) −1537.93 −0.145787
\(482\) −3301.43 −0.311983
\(483\) −1594.17 −0.150181
\(484\) 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
\(490\) 3770.98 0.347664
\(491\) 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
\(497\) −3589.26 −0.323944
\(498\) −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 \(-0.177373\pi\)
0.848721 + 0.528840i \(0.177373\pi\)
\(504\) 666.391 0.0588957
\(505\) 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 \(-0.253256\pi\)
0.699836 + 0.714303i \(0.253256\pi\)
\(510\) −2582.77 −0.224249
\(511\) 4838.58 0.418877
\(512\) 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
\(518\) −1638.09 −0.138945
\(519\) 2421.44 0.204797
\(520\) −817.952 −0.0689799
\(521\) −18955.3 −1.59395 −0.796975 0.604012i \(-0.793568\pi\)
−0.796975 + 0.604012i \(0.793568\pi\)
\(522\) 7378.97 0.618714
\(523\) −4442.19 −0.371402 −0.185701 0.982606i \(-0.559456\pi\)
−0.185701 + 0.982606i \(0.559456\pi\)
\(524\) −3869.40 −0.322587
\(525\) 1605.93 0.133502
\(526\) −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
\(532\) 1259.22 0.102620
\(533\) 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
\(539\) −3525.38 −0.281723
\(540\) −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
\(546\) 721.044 0.0565162
\(547\) 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
\(553\) 1554.08 0.119505
\(554\) 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
\(560\) −1317.83 −0.0994435
\(561\) 2414.56 0.181716
\(562\) −6314.50 −0.473952
\(563\) −4504.50 −0.337197 −0.168599 0.985685i \(-0.553924\pi\)
−0.168599 + 0.985685i \(0.553924\pi\)
\(564\) −1718.54 −0.128304
\(565\) 5685.34 0.423335
\(566\) 7164.36 0.532050
\(567\) −384.310 −0.0284647
\(568\) −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
\(574\) 3860.09 0.280691
\(575\) −12636.5 −0.916485
\(576\) 1373.11 0.0993281
\(577\) 6339.65 0.457406 0.228703 0.973496i \(-0.426552\pi\)
0.228703 + 0.973496i \(0.426552\pi\)
\(578\) 1485.90 0.106929
\(579\) 11843.3 0.850069
\(580\) −2860.68 −0.204799
\(581\) 3589.83 0.256336
\(582\) 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 \(-0.655771\pi\)
−0.470070 + 0.882629i \(0.655771\pi\)
\(588\) −3242.34 −0.227401
\(589\) −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 \(-0.334873\pi\)
0.495804 + 0.868434i \(0.334873\pi\)
\(594\) 1001.57 0.0691832
\(595\) 1211.26 0.0834567
\(596\) 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 \(-0.197628\pi\)
0.813375 + 0.581739i \(0.197628\pi\)
\(602\) 4480.26 0.303325
\(603\) 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 \(-0.788153\pi\)
−0.786585 + 0.617482i \(0.788153\pi\)
\(608\) 11301.9 0.753872
\(609\) −3460.56 −0.230261
\(610\) 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
\(616\) 814.478 0.0532732
\(617\) −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 \(-0.568229\pi\)
−0.212709 + 0.977115i \(0.568229\pi\)
\(620\) −3279.47 −0.212430
\(621\) 3024.00 0.195409
\(622\) 23365.9 1.50625
\(623\) −2414.88 −0.155297
\(624\) 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
\(630\) 502.434 0.0317737
\(631\) −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
\(637\) 4814.31 0.299450
\(638\) 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 \(-0.671568\pi\)
−0.513274 + 0.858225i \(0.671568\pi\)
\(644\) −1792.00 −0.109650
\(645\) 2931.03 0.178929
\(646\) −19419.1 −1.18271
\(647\) 7818.70 0.475092 0.237546 0.971376i \(-0.423657\pi\)
0.237546 + 0.971376i \(0.423657\pi\)
\(648\) −1264.08 −0.0766325
\(649\) 2156.00 0.130401
\(650\) 5715.49 0.344892
\(651\) −3967.17 −0.238842
\(652\) 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
\(658\) 2717.91 0.161026
\(659\) 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 \(-0.460307\pi\)
0.124378 + 0.992235i \(0.460307\pi\)
\(662\) −1825.90 −0.107199
\(663\) −3297.35 −0.193150
\(664\) 11807.8 0.690106
\(665\) −1302.85 −0.0759733
\(666\) 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
\(672\) 2044.04 0.117337
\(673\) 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 \(-0.447932\pi\)
0.162848 + 0.986651i \(0.447932\pi\)
\(678\) −16484.8 −0.933771
\(679\) −2914.86 −0.164745
\(680\) 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
\(686\) 10615.8 0.590837
\(687\) −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 \(-0.649640\pi\)
−0.452983 + 0.891519i \(0.649640\pi\)
\(692\) 2721.93 0.149526
\(693\) −469.712 −0.0257473
\(694\) 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
\(700\) 1805.22 0.0974725
\(701\) 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
\(707\) 4820.09 0.256405
\(708\) 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
\(714\) −3512.09 −0.184085
\(715\) 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 \(-0.750079\pi\)
−0.707282 + 0.706931i \(0.750079\pi\)
\(720\) 2499.80 0.129392
\(721\) −5229.28 −0.270109
\(722\) −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 \(-0.467631\pi\)
0.101514 + 0.994834i \(0.467631\pi\)
\(728\) −1112.26 −0.0566253
\(729\) 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 \(-0.424365\pi\)
0.235384 + 0.971903i \(0.424365\pi\)
\(734\) −28426.3 −1.42947
\(735\) 3354.68 0.168353
\(736\) −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
\(742\) 6554.43 0.324287
\(743\) 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
\(749\) −6536.00 −0.318852
\(750\) 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
\(756\) −432.000 −0.0207827
\(757\) −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 \(-0.560605\pi\)
−0.189248 + 0.981929i \(0.560605\pi\)
\(762\) 23180.3 1.10201
\(763\) −1519.29 −0.0720866
\(764\) 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 \(-0.278817\pi\)
0.640284 + 0.768139i \(0.278817\pi\)
\(770\) 614.086 0.0287404
\(771\) 24390.8 1.13932
\(772\) 13313.0 0.620653
\(773\) −18872.6 −0.878136 −0.439068 0.898454i \(-0.644691\pi\)
−0.439068 + 0.898454i \(0.644691\pi\)
\(774\) −8498.64 −0.394674
\(775\) −31446.6 −1.45754
\(776\) −9587.65 −0.443527
\(777\) −1457.25 −0.0672826
\(778\) −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
\(784\) 25512.8 1.16221
\(785\) 7231.82 0.328808
\(786\) −11608.2 −0.526782
\(787\) −14512.1 −0.657307 −0.328654 0.944450i \(-0.606595\pi\)
−0.328654 + 0.944450i \(0.606595\pi\)
\(788\) 3115.60 0.140849
\(789\) −13648.3 −0.615832
\(790\) 3854.04 0.173570
\(791\) 7731.01 0.347513
\(792\) −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 \(-0.276083\pi\)
0.646856 + 0.762612i \(0.276083\pi\)
\(798\) 3777.65 0.167578
\(799\) −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
\(805\) 1854.09 0.0811777
\(806\) −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 \(-0.456872\pi\)
0.135077 + 0.990835i \(0.456872\pi\)
\(812\) −3890.00 −0.168118
\(813\) 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
\(819\) 641.445 0.0273674
\(820\) 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
\(826\) −3136.00 −0.132101
\(827\) 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 \(-0.565782\pi\)
−0.205192 + 0.978722i \(0.565782\pi\)
\(830\) 8902.62 0.372306
\(831\) 3382.56 0.141203
\(832\) −2291.84 −0.0954990
\(833\) −23449.7 −0.975370
\(834\) −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 \(-0.297510\pi\)
0.594095 + 0.804395i \(0.297510\pi\)
\(840\) −775.041 −0.0318351
\(841\) 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
\(847\) −574.092 −0.0232893
\(848\) 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 \(-0.895350\pi\)
−0.946441 + 0.322878i \(0.895350\pi\)
\(854\) 11221.7 0.449649
\(855\) 2471.38 0.0988531
\(856\) −21498.4 −0.858412
\(857\) 5021.31 0.200145 0.100073 0.994980i \(-0.468092\pi\)
0.100073 + 0.994980i \(0.468092\pi\)
\(858\) −1671.70 −0.0665162
\(859\) −22921.1 −0.910428 −0.455214 0.890382i \(-0.650437\pi\)
−0.455214 + 0.890382i \(0.650437\pi\)
\(860\) 3294.76 0.130640
\(861\) 3433.95 0.135922
\(862\) −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
\(868\) −4459.48 −0.174383
\(869\) −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
\(875\) −3937.06 −0.152111
\(876\) −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 \(-0.569268\pi\)
−0.215897 + 0.976416i \(0.569268\pi\)
\(882\) −9727.02 −0.371344
\(883\) 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 \(-0.392659\pi\)
0.330866 + 0.943678i \(0.392659\pi\)
\(888\) −4793.24 −0.181138
\(889\) −10871.0 −0.410126
\(890\) −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
\(896\) −7891.87 −0.294251
\(897\) −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
\(903\) 3985.66 0.146882
\(904\) 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
\(910\) −838.604 −0.0305489
\(911\) −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
\(917\) 5443.97 0.196048
\(918\) 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
\(924\) −528.000 −0.0187986
\(925\) −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 \(-0.379719\pi\)
0.368945 + 0.929451i \(0.379719\pi\)
\(930\) −9838.42 −0.346897
\(931\) 25222.8 0.887911
\(932\) 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 \(-0.482215\pi\)
0.0558454 + 0.998439i \(0.482215\pi\)
\(938\) −14409.4 −0.501581
\(939\) −10682.2 −0.371248
\(940\) 1998.74 0.0693528
\(941\) 19951.6 0.691182 0.345591 0.938385i \(-0.387678\pi\)
0.345591 + 0.938385i \(0.387678\pi\)
\(942\) −20968.9 −0.725270
\(943\) −27020.6 −0.933099
\(944\) −15602.8 −0.537952
\(945\) 446.968 0.0153861
\(946\) −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
\(952\) 5417.65 0.184440
\(953\) −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
\(959\) −6018.94 −0.202671
\(960\) −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
\(966\) −5376.00 −0.179058
\(967\) 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 \(-0.734526\pi\)
−0.671911 + 0.740632i \(0.734526\pi\)
\(972\) 819.464 0.0270415
\(973\) 2307.23 0.0760188
\(974\) 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
\(980\) 3770.98 0.122918
\(981\) 2881.96 0.0937959
\(982\) 24824.7 0.806709
\(983\) −4435.20 −0.143907 −0.0719536 0.997408i \(-0.522923\pi\)
−0.0719536 + 0.997408i \(0.522923\pi\)
\(984\) 11295.1 0.365929
\(985\) −3223.55 −0.104275
\(986\) 59989.8 1.93759
\(987\) 2417.87 0.0779753
\(988\) 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
\(994\) −12104.0 −0.386233
\(995\) 1661.35 0.0529331
\(996\) −7654.60 −0.243519
\(997\) −53480.1 −1.69883 −0.849413 0.527728i \(-0.823044\pi\)
−0.849413 + 0.527728i \(0.823044\pi\)
\(998\) 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 33.4.a.d.1.2 2
3.2 odd 2 99.4.a.e.1.1 2
4.3 odd 2 528.4.a.o.1.1 2
5.2 odd 4 825.4.c.i.199.4 4
5.3 odd 4 825.4.c.i.199.1 4
5.4 even 2 825.4.a.k.1.1 2
7.6 odd 2 1617.4.a.j.1.2 2
8.3 odd 2 2112.4.a.bh.1.2 2
8.5 even 2 2112.4.a.ba.1.2 2
11.10 odd 2 363.4.a.j.1.1 2
12.11 even 2 1584.4.a.x.1.2 2
15.14 odd 2 2475.4.a.o.1.2 2
33.32 even 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 1.1 even 1 trivial
99.4.a.e.1.1 2 3.2 odd 2
363.4.a.j.1.1 2 11.10 odd 2
528.4.a.o.1.1 2 4.3 odd 2
825.4.a.k.1.1 2 5.4 even 2
825.4.c.i.199.1 4 5.3 odd 4
825.4.c.i.199.4 4 5.2 odd 4
1089.4.a.t.1.2 2 33.32 even 2
1584.4.a.x.1.2 2 12.11 even 2
1617.4.a.j.1.2 2 7.6 odd 2
2112.4.a.ba.1.2 2 8.5 even 2
2112.4.a.bh.1.2 2 8.3 odd 2
2475.4.a.o.1.2 2 15.14 odd 2