Properties

Label 2013.4.a.e.1.4
Level $2013$
Weight $4$
Character 2013.1
Self dual yes
Analytic conductor $118.771$
Analytic rank $0$
Dimension $38$
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] = [2013,4,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(118.770844842\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.79323 q^{2} -3.00000 q^{3} +14.9751 q^{4} -18.8616 q^{5} +14.3797 q^{6} -18.5258 q^{7} -33.4332 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.79323 q^{2} -3.00000 q^{3} +14.9751 q^{4} -18.8616 q^{5} +14.3797 q^{6} -18.5258 q^{7} -33.4332 q^{8} +9.00000 q^{9} +90.4081 q^{10} -11.0000 q^{11} -44.9253 q^{12} +58.9290 q^{13} +88.7987 q^{14} +56.5848 q^{15} +40.4525 q^{16} -109.455 q^{17} -43.1391 q^{18} +9.97788 q^{19} -282.454 q^{20} +55.5775 q^{21} +52.7256 q^{22} +22.2452 q^{23} +100.300 q^{24} +230.760 q^{25} -282.460 q^{26} -27.0000 q^{27} -277.426 q^{28} +23.1129 q^{29} -271.224 q^{30} +40.4720 q^{31} +73.5674 q^{32} +33.0000 q^{33} +524.642 q^{34} +349.427 q^{35} +134.776 q^{36} +96.3869 q^{37} -47.8263 q^{38} -176.787 q^{39} +630.604 q^{40} -78.4205 q^{41} -266.396 q^{42} -29.9808 q^{43} -164.726 q^{44} -169.754 q^{45} -106.626 q^{46} +173.214 q^{47} -121.358 q^{48} +0.206858 q^{49} -1106.09 q^{50} +328.364 q^{51} +882.467 q^{52} -132.685 q^{53} +129.417 q^{54} +207.478 q^{55} +619.378 q^{56} -29.9336 q^{57} -110.786 q^{58} -309.354 q^{59} +847.363 q^{60} -61.0000 q^{61} -193.992 q^{62} -166.733 q^{63} -676.246 q^{64} -1111.50 q^{65} -158.177 q^{66} +103.354 q^{67} -1639.09 q^{68} -66.7355 q^{69} -1674.89 q^{70} -489.246 q^{71} -300.899 q^{72} +79.7769 q^{73} -462.005 q^{74} -692.281 q^{75} +149.420 q^{76} +203.784 q^{77} +847.381 q^{78} -337.338 q^{79} -762.999 q^{80} +81.0000 q^{81} +375.888 q^{82} -640.928 q^{83} +832.278 q^{84} +2064.49 q^{85} +143.705 q^{86} -69.3387 q^{87} +367.765 q^{88} +219.386 q^{89} +813.673 q^{90} -1091.71 q^{91} +333.123 q^{92} -121.416 q^{93} -830.256 q^{94} -188.199 q^{95} -220.702 q^{96} +43.0051 q^{97} -0.991519 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q - 2 q^{2} - 114 q^{3} + 142 q^{4} + 15 q^{5} + 6 q^{6} + 63 q^{7} - 45 q^{8} + 342 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q - 2 q^{2} - 114 q^{3} + 142 q^{4} + 15 q^{5} + 6 q^{6} + 63 q^{7} - 45 q^{8} + 342 q^{9} + 95 q^{10} - 418 q^{11} - 426 q^{12} + 13 q^{13} + 26 q^{14} - 45 q^{15} + 486 q^{16} - 224 q^{17} - 18 q^{18} + 367 q^{19} + 18 q^{20} - 189 q^{21} + 22 q^{22} + 51 q^{23} + 135 q^{24} + 773 q^{25} - 439 q^{26} - 1026 q^{27} + 22 q^{28} - 462 q^{29} - 285 q^{30} + 234 q^{31} - 597 q^{32} + 1254 q^{33} + 956 q^{34} - 522 q^{35} + 1278 q^{36} + 954 q^{37} + 705 q^{38} - 39 q^{39} + 1495 q^{40} - 740 q^{41} - 78 q^{42} + 1441 q^{43} - 1562 q^{44} + 135 q^{45} + 581 q^{46} + 1003 q^{47} - 1458 q^{48} + 2707 q^{49} + 388 q^{50} + 672 q^{51} + 788 q^{52} + 735 q^{53} + 54 q^{54} - 165 q^{55} + 1059 q^{56} - 1101 q^{57} + 177 q^{58} + 261 q^{59} - 54 q^{60} - 2318 q^{61} + 1251 q^{62} + 567 q^{63} + 5571 q^{64} - 1354 q^{65} - 66 q^{66} + 3495 q^{67} - 1856 q^{68} - 153 q^{69} + 542 q^{70} - 873 q^{71} - 405 q^{72} + 989 q^{73} - 3406 q^{74} - 2319 q^{75} + 1712 q^{76} - 693 q^{77} + 1317 q^{78} + 2313 q^{79} + 1593 q^{80} + 3078 q^{81} + 5170 q^{82} + 569 q^{83} - 66 q^{84} - 1271 q^{85} + 3065 q^{86} + 1386 q^{87} + 495 q^{88} - 2917 q^{89} + 855 q^{90} + 2740 q^{91} + 1083 q^{92} - 702 q^{93} + 3272 q^{94} + 2696 q^{95} + 1791 q^{96} + 4250 q^{97} + 5952 q^{98} - 3762 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.79323 −1.69466 −0.847332 0.531064i \(-0.821793\pi\)
−0.847332 + 0.531064i \(0.821793\pi\)
\(3\) −3.00000 −0.577350
\(4\) 14.9751 1.87189
\(5\) −18.8616 −1.68703 −0.843517 0.537103i \(-0.819519\pi\)
−0.843517 + 0.537103i \(0.819519\pi\)
\(6\) 14.3797 0.978415
\(7\) −18.5258 −1.00030 −0.500151 0.865938i \(-0.666722\pi\)
−0.500151 + 0.865938i \(0.666722\pi\)
\(8\) −33.4332 −1.47755
\(9\) 9.00000 0.333333
\(10\) 90.4081 2.85896
\(11\) −11.0000 −0.301511
\(12\) −44.9253 −1.08073
\(13\) 58.9290 1.25723 0.628614 0.777718i \(-0.283623\pi\)
0.628614 + 0.777718i \(0.283623\pi\)
\(14\) 88.7987 1.69517
\(15\) 56.5848 0.974009
\(16\) 40.4525 0.632070
\(17\) −109.455 −1.56157 −0.780784 0.624801i \(-0.785180\pi\)
−0.780784 + 0.624801i \(0.785180\pi\)
\(18\) −43.1391 −0.564888
\(19\) 9.97788 0.120478 0.0602390 0.998184i \(-0.480814\pi\)
0.0602390 + 0.998184i \(0.480814\pi\)
\(20\) −282.454 −3.15793
\(21\) 55.5775 0.577524
\(22\) 52.7256 0.510960
\(23\) 22.2452 0.201671 0.100836 0.994903i \(-0.467848\pi\)
0.100836 + 0.994903i \(0.467848\pi\)
\(24\) 100.300 0.853066
\(25\) 230.760 1.84608
\(26\) −282.460 −2.13058
\(27\) −27.0000 −0.192450
\(28\) −277.426 −1.87245
\(29\) 23.1129 0.147999 0.0739993 0.997258i \(-0.476424\pi\)
0.0739993 + 0.997258i \(0.476424\pi\)
\(30\) −271.224 −1.65062
\(31\) 40.4720 0.234484 0.117242 0.993103i \(-0.462595\pi\)
0.117242 + 0.993103i \(0.462595\pi\)
\(32\) 73.5674 0.406406
\(33\) 33.0000 0.174078
\(34\) 524.642 2.64633
\(35\) 349.427 1.68754
\(36\) 134.776 0.623962
\(37\) 96.3869 0.428268 0.214134 0.976804i \(-0.431307\pi\)
0.214134 + 0.976804i \(0.431307\pi\)
\(38\) −47.8263 −0.204170
\(39\) −176.787 −0.725861
\(40\) 630.604 2.49268
\(41\) −78.4205 −0.298713 −0.149356 0.988783i \(-0.547720\pi\)
−0.149356 + 0.988783i \(0.547720\pi\)
\(42\) −266.396 −0.978710
\(43\) −29.9808 −0.106326 −0.0531631 0.998586i \(-0.516930\pi\)
−0.0531631 + 0.998586i \(0.516930\pi\)
\(44\) −164.726 −0.564395
\(45\) −169.754 −0.562345
\(46\) −106.626 −0.341765
\(47\) 173.214 0.537572 0.268786 0.963200i \(-0.413378\pi\)
0.268786 + 0.963200i \(0.413378\pi\)
\(48\) −121.358 −0.364926
\(49\) 0.206858 0.000603085 0
\(50\) −1106.09 −3.12849
\(51\) 328.364 0.901571
\(52\) 882.467 2.35339
\(53\) −132.685 −0.343880 −0.171940 0.985107i \(-0.555003\pi\)
−0.171940 + 0.985107i \(0.555003\pi\)
\(54\) 129.417 0.326138
\(55\) 207.478 0.508660
\(56\) 619.378 1.47800
\(57\) −29.9336 −0.0695580
\(58\) −110.786 −0.250808
\(59\) −309.354 −0.682618 −0.341309 0.939951i \(-0.610870\pi\)
−0.341309 + 0.939951i \(0.610870\pi\)
\(60\) 847.363 1.82323
\(61\) −61.0000 −0.128037
\(62\) −193.992 −0.397371
\(63\) −166.733 −0.333434
\(64\) −676.246 −1.32079
\(65\) −1111.50 −2.12099
\(66\) −158.177 −0.295003
\(67\) 103.354 0.188458 0.0942290 0.995551i \(-0.469961\pi\)
0.0942290 + 0.995551i \(0.469961\pi\)
\(68\) −1639.09 −2.92308
\(69\) −66.7355 −0.116435
\(70\) −1674.89 −2.85982
\(71\) −489.246 −0.817787 −0.408893 0.912582i \(-0.634085\pi\)
−0.408893 + 0.912582i \(0.634085\pi\)
\(72\) −300.899 −0.492518
\(73\) 79.7769 0.127906 0.0639532 0.997953i \(-0.479629\pi\)
0.0639532 + 0.997953i \(0.479629\pi\)
\(74\) −462.005 −0.725770
\(75\) −692.281 −1.06584
\(76\) 149.420 0.225521
\(77\) 203.784 0.301602
\(78\) 847.381 1.23009
\(79\) −337.338 −0.480424 −0.240212 0.970720i \(-0.577217\pi\)
−0.240212 + 0.970720i \(0.577217\pi\)
\(80\) −762.999 −1.06632
\(81\) 81.0000 0.111111
\(82\) 375.888 0.506218
\(83\) −640.928 −0.847602 −0.423801 0.905755i \(-0.639304\pi\)
−0.423801 + 0.905755i \(0.639304\pi\)
\(84\) 832.278 1.08106
\(85\) 2064.49 2.63442
\(86\) 143.705 0.180187
\(87\) −69.3387 −0.0854470
\(88\) 367.765 0.445499
\(89\) 219.386 0.261291 0.130646 0.991429i \(-0.458295\pi\)
0.130646 + 0.991429i \(0.458295\pi\)
\(90\) 813.673 0.952985
\(91\) −1091.71 −1.25761
\(92\) 333.123 0.377505
\(93\) −121.416 −0.135379
\(94\) −830.256 −0.911003
\(95\) −188.199 −0.203250
\(96\) −220.702 −0.234639
\(97\) 43.0051 0.0450155 0.0225078 0.999747i \(-0.492835\pi\)
0.0225078 + 0.999747i \(0.492835\pi\)
\(98\) −0.991519 −0.00102203
\(99\) −99.0000 −0.100504
\(100\) 3455.66 3.45566
\(101\) −1294.03 −1.27486 −0.637429 0.770509i \(-0.720002\pi\)
−0.637429 + 0.770509i \(0.720002\pi\)
\(102\) −1573.92 −1.52786
\(103\) −691.875 −0.661869 −0.330934 0.943654i \(-0.607364\pi\)
−0.330934 + 0.943654i \(0.607364\pi\)
\(104\) −1970.19 −1.85762
\(105\) −1048.28 −0.974303
\(106\) 635.988 0.582760
\(107\) 976.022 0.881828 0.440914 0.897549i \(-0.354654\pi\)
0.440914 + 0.897549i \(0.354654\pi\)
\(108\) −404.327 −0.360245
\(109\) −863.019 −0.758369 −0.379184 0.925321i \(-0.623795\pi\)
−0.379184 + 0.925321i \(0.623795\pi\)
\(110\) −994.489 −0.862007
\(111\) −289.161 −0.247261
\(112\) −749.417 −0.632261
\(113\) −1057.91 −0.880706 −0.440353 0.897825i \(-0.645147\pi\)
−0.440353 + 0.897825i \(0.645147\pi\)
\(114\) 143.479 0.117877
\(115\) −419.580 −0.340226
\(116\) 346.118 0.277036
\(117\) 530.361 0.419076
\(118\) 1482.80 1.15681
\(119\) 2027.74 1.56204
\(120\) −1891.81 −1.43915
\(121\) 121.000 0.0909091
\(122\) 292.387 0.216979
\(123\) 235.261 0.172462
\(124\) 606.072 0.438926
\(125\) −1994.81 −1.42737
\(126\) 799.188 0.565058
\(127\) 562.287 0.392873 0.196437 0.980517i \(-0.437063\pi\)
0.196437 + 0.980517i \(0.437063\pi\)
\(128\) 2652.86 1.83189
\(129\) 89.9423 0.0613874
\(130\) 5327.66 3.59436
\(131\) 52.1834 0.0348037 0.0174018 0.999849i \(-0.494461\pi\)
0.0174018 + 0.999849i \(0.494461\pi\)
\(132\) 494.178 0.325853
\(133\) −184.849 −0.120514
\(134\) −495.399 −0.319373
\(135\) 509.263 0.324670
\(136\) 3659.42 2.30730
\(137\) 438.281 0.273320 0.136660 0.990618i \(-0.456363\pi\)
0.136660 + 0.990618i \(0.456363\pi\)
\(138\) 319.879 0.197318
\(139\) −989.299 −0.603678 −0.301839 0.953359i \(-0.597600\pi\)
−0.301839 + 0.953359i \(0.597600\pi\)
\(140\) 5232.70 3.15889
\(141\) −519.642 −0.310367
\(142\) 2345.07 1.38587
\(143\) −648.219 −0.379068
\(144\) 364.073 0.210690
\(145\) −435.946 −0.249679
\(146\) −382.389 −0.216759
\(147\) −0.620574 −0.000348191 0
\(148\) 1443.40 0.801669
\(149\) −2484.38 −1.36596 −0.682981 0.730436i \(-0.739317\pi\)
−0.682981 + 0.730436i \(0.739317\pi\)
\(150\) 3318.26 1.80623
\(151\) −2543.02 −1.37052 −0.685258 0.728300i \(-0.740311\pi\)
−0.685258 + 0.728300i \(0.740311\pi\)
\(152\) −333.593 −0.178013
\(153\) −985.092 −0.520522
\(154\) −976.786 −0.511114
\(155\) −763.368 −0.395582
\(156\) −2647.40 −1.35873
\(157\) −136.047 −0.0691576 −0.0345788 0.999402i \(-0.511009\pi\)
−0.0345788 + 0.999402i \(0.511009\pi\)
\(158\) 1616.94 0.814157
\(159\) 398.054 0.198539
\(160\) −1387.60 −0.685621
\(161\) −412.110 −0.201732
\(162\) −388.252 −0.188296
\(163\) −2537.25 −1.21922 −0.609611 0.792701i \(-0.708674\pi\)
−0.609611 + 0.792701i \(0.708674\pi\)
\(164\) −1174.35 −0.559156
\(165\) −622.433 −0.293675
\(166\) 3072.12 1.43640
\(167\) −406.687 −0.188446 −0.0942228 0.995551i \(-0.530037\pi\)
−0.0942228 + 0.995551i \(0.530037\pi\)
\(168\) −1858.14 −0.853323
\(169\) 1275.63 0.580622
\(170\) −9895.58 −4.46445
\(171\) 89.8009 0.0401593
\(172\) −448.964 −0.199030
\(173\) 3353.97 1.47398 0.736988 0.675906i \(-0.236247\pi\)
0.736988 + 0.675906i \(0.236247\pi\)
\(174\) 332.357 0.144804
\(175\) −4275.03 −1.84664
\(176\) −444.978 −0.190576
\(177\) 928.061 0.394109
\(178\) −1051.57 −0.442801
\(179\) −4696.73 −1.96117 −0.980587 0.196087i \(-0.937177\pi\)
−0.980587 + 0.196087i \(0.937177\pi\)
\(180\) −2542.09 −1.05264
\(181\) 1335.68 0.548510 0.274255 0.961657i \(-0.411569\pi\)
0.274255 + 0.961657i \(0.411569\pi\)
\(182\) 5232.82 2.13122
\(183\) 183.000 0.0739221
\(184\) −743.727 −0.297980
\(185\) −1818.01 −0.722503
\(186\) 581.976 0.229422
\(187\) 1204.00 0.470830
\(188\) 2593.90 1.00627
\(189\) 500.198 0.192508
\(190\) 902.081 0.344441
\(191\) −1041.63 −0.394605 −0.197303 0.980343i \(-0.563218\pi\)
−0.197303 + 0.980343i \(0.563218\pi\)
\(192\) 2028.74 0.762560
\(193\) −2512.80 −0.937178 −0.468589 0.883416i \(-0.655237\pi\)
−0.468589 + 0.883416i \(0.655237\pi\)
\(194\) −206.133 −0.0762862
\(195\) 3334.49 1.22455
\(196\) 3.09772 0.00112891
\(197\) −3428.63 −1.24000 −0.620000 0.784602i \(-0.712867\pi\)
−0.620000 + 0.784602i \(0.712867\pi\)
\(198\) 474.530 0.170320
\(199\) −3527.37 −1.25652 −0.628262 0.778002i \(-0.716233\pi\)
−0.628262 + 0.778002i \(0.716233\pi\)
\(200\) −7715.06 −2.72769
\(201\) −310.062 −0.108806
\(202\) 6202.59 2.16046
\(203\) −428.186 −0.148043
\(204\) 4917.28 1.68764
\(205\) 1479.14 0.503938
\(206\) 3316.32 1.12165
\(207\) 200.206 0.0672237
\(208\) 2383.83 0.794656
\(209\) −109.757 −0.0363255
\(210\) 5024.66 1.65112
\(211\) 772.916 0.252179 0.126089 0.992019i \(-0.459757\pi\)
0.126089 + 0.992019i \(0.459757\pi\)
\(212\) −1986.96 −0.643703
\(213\) 1467.74 0.472149
\(214\) −4678.30 −1.49440
\(215\) 565.485 0.179376
\(216\) 902.697 0.284355
\(217\) −749.778 −0.234554
\(218\) 4136.65 1.28518
\(219\) −239.331 −0.0738468
\(220\) 3107.00 0.952153
\(221\) −6450.05 −1.96325
\(222\) 1386.02 0.419024
\(223\) 5306.49 1.59349 0.796746 0.604314i \(-0.206553\pi\)
0.796746 + 0.604314i \(0.206553\pi\)
\(224\) −1362.90 −0.406529
\(225\) 2076.84 0.615361
\(226\) 5070.81 1.49250
\(227\) −1625.41 −0.475251 −0.237626 0.971357i \(-0.576369\pi\)
−0.237626 + 0.971357i \(0.576369\pi\)
\(228\) −448.259 −0.130205
\(229\) −2780.64 −0.802402 −0.401201 0.915990i \(-0.631407\pi\)
−0.401201 + 0.915990i \(0.631407\pi\)
\(230\) 2011.14 0.576569
\(231\) −611.353 −0.174130
\(232\) −772.738 −0.218676
\(233\) −4367.06 −1.22788 −0.613939 0.789353i \(-0.710416\pi\)
−0.613939 + 0.789353i \(0.710416\pi\)
\(234\) −2542.14 −0.710193
\(235\) −3267.10 −0.906902
\(236\) −4632.60 −1.27778
\(237\) 1012.01 0.277373
\(238\) −9719.43 −2.64713
\(239\) 1421.27 0.384663 0.192331 0.981330i \(-0.438395\pi\)
0.192331 + 0.981330i \(0.438395\pi\)
\(240\) 2289.00 0.615642
\(241\) 910.885 0.243466 0.121733 0.992563i \(-0.461155\pi\)
0.121733 + 0.992563i \(0.461155\pi\)
\(242\) −579.981 −0.154060
\(243\) −243.000 −0.0641500
\(244\) −913.480 −0.239670
\(245\) −3.90168 −0.00101742
\(246\) −1127.66 −0.292265
\(247\) 587.986 0.151468
\(248\) −1353.11 −0.346462
\(249\) 1922.78 0.489363
\(250\) 9561.59 2.41891
\(251\) −3620.98 −0.910574 −0.455287 0.890345i \(-0.650463\pi\)
−0.455287 + 0.890345i \(0.650463\pi\)
\(252\) −2496.83 −0.624150
\(253\) −244.697 −0.0608061
\(254\) −2695.17 −0.665788
\(255\) −6193.47 −1.52098
\(256\) −7305.83 −1.78365
\(257\) −7784.06 −1.88932 −0.944662 0.328046i \(-0.893610\pi\)
−0.944662 + 0.328046i \(0.893610\pi\)
\(258\) −431.114 −0.104031
\(259\) −1785.65 −0.428397
\(260\) −16644.7 −3.97024
\(261\) 208.016 0.0493328
\(262\) −250.127 −0.0589805
\(263\) 692.609 0.162388 0.0811941 0.996698i \(-0.474127\pi\)
0.0811941 + 0.996698i \(0.474127\pi\)
\(264\) −1103.30 −0.257209
\(265\) 2502.64 0.580136
\(266\) 886.023 0.204231
\(267\) −658.159 −0.150856
\(268\) 1547.73 0.352772
\(269\) −4169.68 −0.945093 −0.472547 0.881306i \(-0.656665\pi\)
−0.472547 + 0.881306i \(0.656665\pi\)
\(270\) −2441.02 −0.550206
\(271\) 5151.66 1.15476 0.577382 0.816474i \(-0.304074\pi\)
0.577382 + 0.816474i \(0.304074\pi\)
\(272\) −4427.71 −0.987020
\(273\) 3275.13 0.726080
\(274\) −2100.78 −0.463186
\(275\) −2538.36 −0.556615
\(276\) −999.369 −0.217953
\(277\) 1033.41 0.224158 0.112079 0.993699i \(-0.464249\pi\)
0.112079 + 0.993699i \(0.464249\pi\)
\(278\) 4741.94 1.02303
\(279\) 364.248 0.0781612
\(280\) −11682.5 −2.49343
\(281\) 6683.56 1.41889 0.709444 0.704762i \(-0.248946\pi\)
0.709444 + 0.704762i \(0.248946\pi\)
\(282\) 2490.77 0.525968
\(283\) 7916.57 1.66287 0.831434 0.555624i \(-0.187521\pi\)
0.831434 + 0.555624i \(0.187521\pi\)
\(284\) −7326.51 −1.53080
\(285\) 564.597 0.117347
\(286\) 3107.06 0.642394
\(287\) 1452.81 0.298803
\(288\) 662.107 0.135469
\(289\) 7067.31 1.43849
\(290\) 2089.59 0.423121
\(291\) −129.015 −0.0259897
\(292\) 1194.67 0.239426
\(293\) −6375.73 −1.27124 −0.635622 0.772001i \(-0.719256\pi\)
−0.635622 + 0.772001i \(0.719256\pi\)
\(294\) 2.97456 0.000590067 0
\(295\) 5834.91 1.15160
\(296\) −3222.53 −0.632789
\(297\) 297.000 0.0580259
\(298\) 11908.2 2.31485
\(299\) 1310.88 0.253547
\(300\) −10367.0 −1.99512
\(301\) 555.419 0.106358
\(302\) 12189.3 2.32256
\(303\) 3882.09 0.736040
\(304\) 403.630 0.0761506
\(305\) 1150.56 0.216003
\(306\) 4721.77 0.882111
\(307\) −4931.49 −0.916792 −0.458396 0.888748i \(-0.651576\pi\)
−0.458396 + 0.888748i \(0.651576\pi\)
\(308\) 3051.69 0.564565
\(309\) 2075.63 0.382130
\(310\) 3659.00 0.670378
\(311\) 4864.56 0.886958 0.443479 0.896285i \(-0.353744\pi\)
0.443479 + 0.896285i \(0.353744\pi\)
\(312\) 5910.56 1.07250
\(313\) 6219.53 1.12316 0.561579 0.827423i \(-0.310194\pi\)
0.561579 + 0.827423i \(0.310194\pi\)
\(314\) 652.106 0.117199
\(315\) 3144.84 0.562514
\(316\) −5051.66 −0.899298
\(317\) 7094.59 1.25701 0.628505 0.777806i \(-0.283667\pi\)
0.628505 + 0.777806i \(0.283667\pi\)
\(318\) −1907.96 −0.336457
\(319\) −254.242 −0.0446232
\(320\) 12755.1 2.22822
\(321\) −2928.07 −0.509124
\(322\) 1975.34 0.341868
\(323\) −1092.12 −0.188135
\(324\) 1212.98 0.207987
\(325\) 13598.5 2.32095
\(326\) 12161.7 2.06617
\(327\) 2589.06 0.437844
\(328\) 2621.85 0.441364
\(329\) −3208.94 −0.537734
\(330\) 2983.47 0.497680
\(331\) 6.12699 0.00101743 0.000508715 1.00000i \(-0.499838\pi\)
0.000508715 1.00000i \(0.499838\pi\)
\(332\) −9597.95 −1.58661
\(333\) 867.483 0.142756
\(334\) 1949.35 0.319352
\(335\) −1949.42 −0.317935
\(336\) 2248.25 0.365036
\(337\) −9683.53 −1.56527 −0.782634 0.622482i \(-0.786124\pi\)
−0.782634 + 0.622482i \(0.786124\pi\)
\(338\) −6114.37 −0.983959
\(339\) 3173.73 0.508476
\(340\) 30915.9 4.93133
\(341\) −445.192 −0.0706995
\(342\) −430.437 −0.0680566
\(343\) 6350.53 0.999698
\(344\) 1002.35 0.157102
\(345\) 1258.74 0.196430
\(346\) −16076.4 −2.49789
\(347\) 7814.64 1.20897 0.604484 0.796617i \(-0.293379\pi\)
0.604484 + 0.796617i \(0.293379\pi\)
\(348\) −1038.35 −0.159947
\(349\) −388.966 −0.0596587 −0.0298293 0.999555i \(-0.509496\pi\)
−0.0298293 + 0.999555i \(0.509496\pi\)
\(350\) 20491.2 3.12943
\(351\) −1591.08 −0.241954
\(352\) −809.242 −0.122536
\(353\) −7425.41 −1.11959 −0.559794 0.828632i \(-0.689120\pi\)
−0.559794 + 0.828632i \(0.689120\pi\)
\(354\) −4448.41 −0.667883
\(355\) 9227.97 1.37963
\(356\) 3285.33 0.489107
\(357\) −6083.22 −0.901843
\(358\) 22512.5 3.32353
\(359\) 1652.68 0.242967 0.121484 0.992593i \(-0.461235\pi\)
0.121484 + 0.992593i \(0.461235\pi\)
\(360\) 5675.44 0.830894
\(361\) −6759.44 −0.985485
\(362\) −6402.23 −0.929541
\(363\) −363.000 −0.0524864
\(364\) −16348.4 −2.35410
\(365\) −1504.72 −0.215783
\(366\) −877.162 −0.125273
\(367\) 4307.76 0.612706 0.306353 0.951918i \(-0.400891\pi\)
0.306353 + 0.951918i \(0.400891\pi\)
\(368\) 899.872 0.127470
\(369\) −705.784 −0.0995709
\(370\) 8714.16 1.22440
\(371\) 2458.09 0.343983
\(372\) −1818.22 −0.253414
\(373\) 4742.18 0.658286 0.329143 0.944280i \(-0.393240\pi\)
0.329143 + 0.944280i \(0.393240\pi\)
\(374\) −5771.06 −0.797899
\(375\) 5984.43 0.824092
\(376\) −5791.10 −0.794291
\(377\) 1362.02 0.186068
\(378\) −2397.56 −0.326237
\(379\) −8767.91 −1.18833 −0.594165 0.804343i \(-0.702518\pi\)
−0.594165 + 0.804343i \(0.702518\pi\)
\(380\) −2818.29 −0.380462
\(381\) −1686.86 −0.226825
\(382\) 4992.77 0.668724
\(383\) 11781.3 1.57179 0.785895 0.618360i \(-0.212202\pi\)
0.785895 + 0.618360i \(0.212202\pi\)
\(384\) −7958.59 −1.05764
\(385\) −3843.70 −0.508813
\(386\) 12044.4 1.58820
\(387\) −269.827 −0.0354420
\(388\) 644.005 0.0842639
\(389\) 1332.13 0.173629 0.0868143 0.996225i \(-0.472331\pi\)
0.0868143 + 0.996225i \(0.472331\pi\)
\(390\) −15983.0 −2.07520
\(391\) −2434.84 −0.314923
\(392\) −6.91593 −0.000891090 0
\(393\) −156.550 −0.0200939
\(394\) 16434.2 2.10138
\(395\) 6362.74 0.810491
\(396\) −1482.53 −0.188132
\(397\) −11017.4 −1.39282 −0.696409 0.717645i \(-0.745220\pi\)
−0.696409 + 0.717645i \(0.745220\pi\)
\(398\) 16907.5 2.12939
\(399\) 554.546 0.0695790
\(400\) 9334.83 1.16685
\(401\) 13120.1 1.63388 0.816940 0.576722i \(-0.195669\pi\)
0.816940 + 0.576722i \(0.195669\pi\)
\(402\) 1486.20 0.184390
\(403\) 2384.98 0.294799
\(404\) −19378.2 −2.38639
\(405\) −1527.79 −0.187448
\(406\) 2052.39 0.250883
\(407\) −1060.26 −0.129128
\(408\) −10978.3 −1.33212
\(409\) −4856.51 −0.587137 −0.293569 0.955938i \(-0.594843\pi\)
−0.293569 + 0.955938i \(0.594843\pi\)
\(410\) −7089.85 −0.854006
\(411\) −1314.84 −0.157801
\(412\) −10360.9 −1.23894
\(413\) 5731.04 0.682823
\(414\) −959.636 −0.113922
\(415\) 12088.9 1.42993
\(416\) 4335.25 0.510945
\(417\) 2967.90 0.348534
\(418\) 526.089 0.0615595
\(419\) 4568.10 0.532617 0.266308 0.963888i \(-0.414196\pi\)
0.266308 + 0.963888i \(0.414196\pi\)
\(420\) −15698.1 −1.82378
\(421\) 5706.02 0.660557 0.330278 0.943884i \(-0.392857\pi\)
0.330278 + 0.943884i \(0.392857\pi\)
\(422\) −3704.76 −0.427358
\(423\) 1558.93 0.179191
\(424\) 4436.07 0.508100
\(425\) −25257.8 −2.88278
\(426\) −7035.22 −0.800134
\(427\) 1130.08 0.128075
\(428\) 14616.0 1.65068
\(429\) 1944.66 0.218855
\(430\) −2710.50 −0.303982
\(431\) 312.058 0.0348754 0.0174377 0.999848i \(-0.494449\pi\)
0.0174377 + 0.999848i \(0.494449\pi\)
\(432\) −1092.22 −0.121642
\(433\) 15365.2 1.70533 0.852664 0.522460i \(-0.174986\pi\)
0.852664 + 0.522460i \(0.174986\pi\)
\(434\) 3593.86 0.397491
\(435\) 1307.84 0.144152
\(436\) −12923.8 −1.41958
\(437\) 221.960 0.0242969
\(438\) 1147.17 0.125146
\(439\) −13092.4 −1.42339 −0.711693 0.702491i \(-0.752071\pi\)
−0.711693 + 0.702491i \(0.752071\pi\)
\(440\) −6936.65 −0.751572
\(441\) 1.86172 0.000201028 0
\(442\) 30916.6 3.32704
\(443\) 7799.41 0.836481 0.418241 0.908336i \(-0.362647\pi\)
0.418241 + 0.908336i \(0.362647\pi\)
\(444\) −4330.21 −0.462844
\(445\) −4137.98 −0.440807
\(446\) −25435.2 −2.70043
\(447\) 7453.14 0.788639
\(448\) 12528.0 1.32119
\(449\) 16633.3 1.74828 0.874138 0.485677i \(-0.161427\pi\)
0.874138 + 0.485677i \(0.161427\pi\)
\(450\) −9954.79 −1.04283
\(451\) 862.625 0.0900653
\(452\) −15842.3 −1.64858
\(453\) 7629.06 0.791268
\(454\) 7790.95 0.805391
\(455\) 20591.4 2.12163
\(456\) 1000.78 0.102776
\(457\) −5873.50 −0.601205 −0.300603 0.953749i \(-0.597188\pi\)
−0.300603 + 0.953749i \(0.597188\pi\)
\(458\) 13328.3 1.35980
\(459\) 2955.27 0.300524
\(460\) −6283.24 −0.636864
\(461\) −3848.23 −0.388785 −0.194393 0.980924i \(-0.562274\pi\)
−0.194393 + 0.980924i \(0.562274\pi\)
\(462\) 2930.36 0.295092
\(463\) −12000.7 −1.20458 −0.602291 0.798277i \(-0.705745\pi\)
−0.602291 + 0.798277i \(0.705745\pi\)
\(464\) 934.975 0.0935455
\(465\) 2290.10 0.228389
\(466\) 20932.3 2.08084
\(467\) −5908.05 −0.585422 −0.292711 0.956201i \(-0.594557\pi\)
−0.292711 + 0.956201i \(0.594557\pi\)
\(468\) 7942.20 0.784462
\(469\) −1914.72 −0.188515
\(470\) 15660.0 1.53689
\(471\) 408.141 0.0399282
\(472\) 10342.7 1.00860
\(473\) 329.788 0.0320585
\(474\) −4850.82 −0.470054
\(475\) 2302.50 0.222412
\(476\) 30365.6 2.92396
\(477\) −1194.16 −0.114627
\(478\) −6812.48 −0.651874
\(479\) −18102.6 −1.72679 −0.863393 0.504531i \(-0.831665\pi\)
−0.863393 + 0.504531i \(0.831665\pi\)
\(480\) 4162.80 0.395844
\(481\) 5679.99 0.538430
\(482\) −4366.09 −0.412593
\(483\) 1236.33 0.116470
\(484\) 1811.99 0.170171
\(485\) −811.145 −0.0759427
\(486\) 1164.76 0.108713
\(487\) −7731.38 −0.719388 −0.359694 0.933070i \(-0.617119\pi\)
−0.359694 + 0.933070i \(0.617119\pi\)
\(488\) 2039.43 0.189181
\(489\) 7611.76 0.703918
\(490\) 18.7016 0.00172419
\(491\) −11164.5 −1.02616 −0.513080 0.858341i \(-0.671496\pi\)
−0.513080 + 0.858341i \(0.671496\pi\)
\(492\) 3523.06 0.322829
\(493\) −2529.81 −0.231110
\(494\) −2818.36 −0.256688
\(495\) 1867.30 0.169553
\(496\) 1637.19 0.148210
\(497\) 9063.70 0.818033
\(498\) −9216.35 −0.829306
\(499\) 4923.93 0.441734 0.220867 0.975304i \(-0.429111\pi\)
0.220867 + 0.975304i \(0.429111\pi\)
\(500\) −29872.4 −2.67187
\(501\) 1220.06 0.108799
\(502\) 17356.2 1.54312
\(503\) 16329.3 1.44749 0.723744 0.690068i \(-0.242420\pi\)
0.723744 + 0.690068i \(0.242420\pi\)
\(504\) 5574.41 0.492666
\(505\) 24407.5 2.15073
\(506\) 1172.89 0.103046
\(507\) −3826.88 −0.335222
\(508\) 8420.29 0.735414
\(509\) 2997.63 0.261037 0.130518 0.991446i \(-0.458336\pi\)
0.130518 + 0.991446i \(0.458336\pi\)
\(510\) 29686.8 2.57755
\(511\) −1477.93 −0.127945
\(512\) 13795.6 1.19080
\(513\) −269.403 −0.0231860
\(514\) 37310.8 3.20177
\(515\) 13049.9 1.11659
\(516\) 1346.89 0.114910
\(517\) −1905.35 −0.162084
\(518\) 8559.03 0.725989
\(519\) −10061.9 −0.851001
\(520\) 37160.9 3.13387
\(521\) 12999.6 1.09313 0.546567 0.837416i \(-0.315935\pi\)
0.546567 + 0.837416i \(0.315935\pi\)
\(522\) −997.070 −0.0836026
\(523\) 747.484 0.0624956 0.0312478 0.999512i \(-0.490052\pi\)
0.0312478 + 0.999512i \(0.490052\pi\)
\(524\) 781.450 0.0651485
\(525\) 12825.1 1.06616
\(526\) −3319.84 −0.275193
\(527\) −4429.85 −0.366162
\(528\) 1334.93 0.110029
\(529\) −11672.2 −0.959329
\(530\) −11995.8 −0.983136
\(531\) −2784.18 −0.227539
\(532\) −2768.12 −0.225589
\(533\) −4621.24 −0.375550
\(534\) 3154.71 0.255651
\(535\) −18409.3 −1.48767
\(536\) −3455.45 −0.278457
\(537\) 14090.2 1.13228
\(538\) 19986.3 1.60162
\(539\) −2.27544 −0.000181837 0
\(540\) 7626.26 0.607745
\(541\) 15587.5 1.23874 0.619370 0.785099i \(-0.287388\pi\)
0.619370 + 0.785099i \(0.287388\pi\)
\(542\) −24693.1 −1.95694
\(543\) −4007.04 −0.316683
\(544\) −8052.29 −0.634631
\(545\) 16277.9 1.27939
\(546\) −15698.5 −1.23046
\(547\) −1789.45 −0.139874 −0.0699371 0.997551i \(-0.522280\pi\)
−0.0699371 + 0.997551i \(0.522280\pi\)
\(548\) 6563.29 0.511624
\(549\) −549.000 −0.0426790
\(550\) 12167.0 0.943275
\(551\) 230.618 0.0178306
\(552\) 2231.18 0.172039
\(553\) 6249.47 0.480569
\(554\) −4953.38 −0.379872
\(555\) 5454.04 0.417137
\(556\) −14814.8 −1.13002
\(557\) 7793.66 0.592869 0.296434 0.955053i \(-0.404202\pi\)
0.296434 + 0.955053i \(0.404202\pi\)
\(558\) −1745.93 −0.132457
\(559\) −1766.74 −0.133676
\(560\) 14135.2 1.06665
\(561\) −3612.00 −0.271834
\(562\) −32035.9 −2.40454
\(563\) −1845.05 −0.138116 −0.0690581 0.997613i \(-0.521999\pi\)
−0.0690581 + 0.997613i \(0.521999\pi\)
\(564\) −7781.69 −0.580972
\(565\) 19953.9 1.48578
\(566\) −37946.0 −2.81800
\(567\) −1500.59 −0.111145
\(568\) 16357.1 1.20832
\(569\) −8875.38 −0.653911 −0.326955 0.945040i \(-0.606023\pi\)
−0.326955 + 0.945040i \(0.606023\pi\)
\(570\) −2706.24 −0.198863
\(571\) −8181.61 −0.599632 −0.299816 0.953997i \(-0.596925\pi\)
−0.299816 + 0.953997i \(0.596925\pi\)
\(572\) −9707.13 −0.709573
\(573\) 3124.89 0.227826
\(574\) −6963.64 −0.506370
\(575\) 5133.30 0.372302
\(576\) −6086.21 −0.440264
\(577\) 25816.5 1.86266 0.931330 0.364176i \(-0.118650\pi\)
0.931330 + 0.364176i \(0.118650\pi\)
\(578\) −33875.3 −2.43776
\(579\) 7538.40 0.541080
\(580\) −6528.34 −0.467370
\(581\) 11873.7 0.847857
\(582\) 618.400 0.0440438
\(583\) 1459.53 0.103684
\(584\) −2667.20 −0.188989
\(585\) −10003.5 −0.706995
\(586\) 30560.4 2.15433
\(587\) −8100.43 −0.569575 −0.284788 0.958591i \(-0.591923\pi\)
−0.284788 + 0.958591i \(0.591923\pi\)
\(588\) −9.29315 −0.000651774 0
\(589\) 403.825 0.0282501
\(590\) −27968.1 −1.95157
\(591\) 10285.9 0.715914
\(592\) 3899.09 0.270696
\(593\) −156.409 −0.0108313 −0.00541564 0.999985i \(-0.501724\pi\)
−0.00541564 + 0.999985i \(0.501724\pi\)
\(594\) −1423.59 −0.0983344
\(595\) −38246.4 −2.63521
\(596\) −37203.8 −2.55693
\(597\) 10582.1 0.725455
\(598\) −6283.38 −0.429676
\(599\) −6206.03 −0.423325 −0.211662 0.977343i \(-0.567888\pi\)
−0.211662 + 0.977343i \(0.567888\pi\)
\(600\) 23145.2 1.57483
\(601\) −1228.88 −0.0834058 −0.0417029 0.999130i \(-0.513278\pi\)
−0.0417029 + 0.999130i \(0.513278\pi\)
\(602\) −2662.25 −0.180241
\(603\) 930.185 0.0628193
\(604\) −38081.9 −2.56545
\(605\) −2282.25 −0.153367
\(606\) −18607.8 −1.24734
\(607\) 13556.4 0.906484 0.453242 0.891388i \(-0.350267\pi\)
0.453242 + 0.891388i \(0.350267\pi\)
\(608\) 734.047 0.0489630
\(609\) 1284.56 0.0854728
\(610\) −5514.89 −0.366052
\(611\) 10207.3 0.675850
\(612\) −14751.8 −0.974359
\(613\) 3560.47 0.234594 0.117297 0.993097i \(-0.462577\pi\)
0.117297 + 0.993097i \(0.462577\pi\)
\(614\) 23637.8 1.55365
\(615\) −4437.41 −0.290949
\(616\) −6813.16 −0.445633
\(617\) 10574.1 0.689949 0.344974 0.938612i \(-0.387888\pi\)
0.344974 + 0.938612i \(0.387888\pi\)
\(618\) −9948.96 −0.647582
\(619\) −8415.44 −0.546438 −0.273219 0.961952i \(-0.588088\pi\)
−0.273219 + 0.961952i \(0.588088\pi\)
\(620\) −11431.5 −0.740484
\(621\) −600.619 −0.0388116
\(622\) −23317.0 −1.50310
\(623\) −4064.32 −0.261370
\(624\) −7151.48 −0.458795
\(625\) 8780.29 0.561938
\(626\) −29811.6 −1.90338
\(627\) 329.270 0.0209725
\(628\) −2037.32 −0.129455
\(629\) −10550.0 −0.668769
\(630\) −15074.0 −0.953272
\(631\) −5319.98 −0.335634 −0.167817 0.985818i \(-0.553672\pi\)
−0.167817 + 0.985818i \(0.553672\pi\)
\(632\) 11278.3 0.709852
\(633\) −2318.75 −0.145595
\(634\) −34006.0 −2.13021
\(635\) −10605.6 −0.662790
\(636\) 5960.89 0.371642
\(637\) 12.1899 0.000758215 0
\(638\) 1218.64 0.0756214
\(639\) −4403.22 −0.272596
\(640\) −50037.3 −3.09047
\(641\) −4860.98 −0.299528 −0.149764 0.988722i \(-0.547851\pi\)
−0.149764 + 0.988722i \(0.547851\pi\)
\(642\) 14034.9 0.862793
\(643\) 14243.1 0.873552 0.436776 0.899570i \(-0.356120\pi\)
0.436776 + 0.899570i \(0.356120\pi\)
\(644\) −6171.39 −0.377619
\(645\) −1696.46 −0.103563
\(646\) 5234.81 0.318825
\(647\) −3965.55 −0.240961 −0.120481 0.992716i \(-0.538444\pi\)
−0.120481 + 0.992716i \(0.538444\pi\)
\(648\) −2708.09 −0.164173
\(649\) 3402.89 0.205817
\(650\) −65180.7 −3.93322
\(651\) 2249.34 0.135420
\(652\) −37995.6 −2.28224
\(653\) −10569.8 −0.633430 −0.316715 0.948521i \(-0.602580\pi\)
−0.316715 + 0.948521i \(0.602580\pi\)
\(654\) −12409.9 −0.741999
\(655\) −984.262 −0.0587150
\(656\) −3172.30 −0.188807
\(657\) 717.992 0.0426355
\(658\) 15381.2 0.911278
\(659\) 10893.7 0.643946 0.321973 0.946749i \(-0.395654\pi\)
0.321973 + 0.946749i \(0.395654\pi\)
\(660\) −9320.99 −0.549726
\(661\) 7773.81 0.457437 0.228719 0.973493i \(-0.426546\pi\)
0.228719 + 0.973493i \(0.426546\pi\)
\(662\) −29.3681 −0.00172420
\(663\) 19350.2 1.13348
\(664\) 21428.3 1.25238
\(665\) 3486.54 0.203312
\(666\) −4158.05 −0.241923
\(667\) 514.150 0.0298470
\(668\) −6090.18 −0.352749
\(669\) −15919.5 −0.920004
\(670\) 9344.03 0.538793
\(671\) 671.000 0.0386046
\(672\) 4088.69 0.234710
\(673\) −14506.9 −0.830907 −0.415454 0.909614i \(-0.636377\pi\)
−0.415454 + 0.909614i \(0.636377\pi\)
\(674\) 46415.4 2.65260
\(675\) −6230.53 −0.355279
\(676\) 19102.6 1.08686
\(677\) −8803.40 −0.499767 −0.249883 0.968276i \(-0.580392\pi\)
−0.249883 + 0.968276i \(0.580392\pi\)
\(678\) −15212.4 −0.861695
\(679\) −796.706 −0.0450291
\(680\) −69022.5 −3.89249
\(681\) 4876.22 0.274386
\(682\) 2133.91 0.119812
\(683\) 10311.0 0.577657 0.288829 0.957381i \(-0.406734\pi\)
0.288829 + 0.957381i \(0.406734\pi\)
\(684\) 1344.78 0.0751737
\(685\) −8266.68 −0.461100
\(686\) −30439.6 −1.69415
\(687\) 8341.93 0.463267
\(688\) −1212.80 −0.0672056
\(689\) −7818.96 −0.432335
\(690\) −6033.43 −0.332882
\(691\) 21715.5 1.19551 0.597755 0.801679i \(-0.296059\pi\)
0.597755 + 0.801679i \(0.296059\pi\)
\(692\) 50226.0 2.75912
\(693\) 1834.06 0.100534
\(694\) −37457.4 −2.04879
\(695\) 18659.8 1.01842
\(696\) 2318.22 0.126252
\(697\) 8583.48 0.466460
\(698\) 1864.41 0.101101
\(699\) 13101.2 0.708916
\(700\) −64018.9 −3.45670
\(701\) −20905.0 −1.12635 −0.563175 0.826338i \(-0.690420\pi\)
−0.563175 + 0.826338i \(0.690420\pi\)
\(702\) 7626.43 0.410030
\(703\) 961.737 0.0515969
\(704\) 7438.70 0.398234
\(705\) 9801.29 0.523600
\(706\) 35591.7 1.89733
\(707\) 23973.0 1.27524
\(708\) 13897.8 0.737728
\(709\) 35474.1 1.87907 0.939533 0.342459i \(-0.111260\pi\)
0.939533 + 0.342459i \(0.111260\pi\)
\(710\) −44231.8 −2.33802
\(711\) −3036.04 −0.160141
\(712\) −7334.79 −0.386071
\(713\) 900.307 0.0472886
\(714\) 29158.3 1.52832
\(715\) 12226.5 0.639501
\(716\) −70333.9 −3.67109
\(717\) −4263.81 −0.222085
\(718\) −7921.70 −0.411748
\(719\) 35219.3 1.82679 0.913393 0.407078i \(-0.133452\pi\)
0.913393 + 0.407078i \(0.133452\pi\)
\(720\) −6866.99 −0.355441
\(721\) 12817.6 0.662068
\(722\) 32399.6 1.67007
\(723\) −2732.66 −0.140565
\(724\) 20001.9 1.02675
\(725\) 5333.54 0.273218
\(726\) 1739.94 0.0889468
\(727\) 24680.5 1.25908 0.629538 0.776970i \(-0.283244\pi\)
0.629538 + 0.776970i \(0.283244\pi\)
\(728\) 36499.3 1.85818
\(729\) 729.000 0.0370370
\(730\) 7212.47 0.365679
\(731\) 3281.53 0.166035
\(732\) 2740.44 0.138374
\(733\) −30881.4 −1.55611 −0.778055 0.628196i \(-0.783794\pi\)
−0.778055 + 0.628196i \(0.783794\pi\)
\(734\) −20648.1 −1.03833
\(735\) 11.7050 0.000587410 0
\(736\) 1636.52 0.0819604
\(737\) −1136.89 −0.0568222
\(738\) 3382.99 0.168739
\(739\) −26058.9 −1.29715 −0.648575 0.761151i \(-0.724635\pi\)
−0.648575 + 0.761151i \(0.724635\pi\)
\(740\) −27224.9 −1.35244
\(741\) −1763.96 −0.0874503
\(742\) −11782.2 −0.582936
\(743\) −4982.23 −0.246003 −0.123001 0.992406i \(-0.539252\pi\)
−0.123001 + 0.992406i \(0.539252\pi\)
\(744\) 4059.33 0.200030
\(745\) 46859.4 2.30443
\(746\) −22730.4 −1.11557
\(747\) −5768.35 −0.282534
\(748\) 18030.0 0.881340
\(749\) −18081.6 −0.882094
\(750\) −28684.8 −1.39656
\(751\) −13658.8 −0.663671 −0.331835 0.943337i \(-0.607668\pi\)
−0.331835 + 0.943337i \(0.607668\pi\)
\(752\) 7006.94 0.339783
\(753\) 10862.9 0.525720
\(754\) −6528.48 −0.315323
\(755\) 47965.4 2.31211
\(756\) 7490.50 0.360353
\(757\) −22713.8 −1.09055 −0.545275 0.838257i \(-0.683575\pi\)
−0.545275 + 0.838257i \(0.683575\pi\)
\(758\) 42026.6 2.01382
\(759\) 734.090 0.0351064
\(760\) 6292.09 0.300313
\(761\) −34502.3 −1.64351 −0.821754 0.569843i \(-0.807004\pi\)
−0.821754 + 0.569843i \(0.807004\pi\)
\(762\) 8085.52 0.384393
\(763\) 15988.1 0.758597
\(764\) −15598.5 −0.738656
\(765\) 18580.4 0.878139
\(766\) −56470.5 −2.66366
\(767\) −18229.9 −0.858206
\(768\) 21917.5 1.02979
\(769\) −24960.0 −1.17046 −0.585229 0.810868i \(-0.698995\pi\)
−0.585229 + 0.810868i \(0.698995\pi\)
\(770\) 18423.7 0.862267
\(771\) 23352.2 1.09080
\(772\) −37629.4 −1.75429
\(773\) 26785.9 1.24634 0.623172 0.782085i \(-0.285844\pi\)
0.623172 + 0.782085i \(0.285844\pi\)
\(774\) 1293.34 0.0600623
\(775\) 9339.34 0.432876
\(776\) −1437.80 −0.0665128
\(777\) 5356.95 0.247335
\(778\) −6385.20 −0.294242
\(779\) −782.470 −0.0359883
\(780\) 49934.2 2.29222
\(781\) 5381.71 0.246572
\(782\) 11670.7 0.533689
\(783\) −624.048 −0.0284823
\(784\) 8.36793 0.000381192 0
\(785\) 2566.07 0.116671
\(786\) 750.381 0.0340524
\(787\) −11184.1 −0.506570 −0.253285 0.967392i \(-0.581511\pi\)
−0.253285 + 0.967392i \(0.581511\pi\)
\(788\) −51344.1 −2.32114
\(789\) −2077.83 −0.0937549
\(790\) −30498.1 −1.37351
\(791\) 19598.7 0.880971
\(792\) 3309.89 0.148500
\(793\) −3594.67 −0.160972
\(794\) 52809.1 2.36036
\(795\) −7507.93 −0.334942
\(796\) −52822.6 −2.35207
\(797\) 991.876 0.0440829 0.0220414 0.999757i \(-0.492983\pi\)
0.0220414 + 0.999757i \(0.492983\pi\)
\(798\) −2658.07 −0.117913
\(799\) −18959.1 −0.839454
\(800\) 16976.4 0.750260
\(801\) 1974.48 0.0870970
\(802\) −62887.6 −2.76888
\(803\) −877.545 −0.0385653
\(804\) −4643.20 −0.203673
\(805\) 7773.06 0.340329
\(806\) −11431.7 −0.499586
\(807\) 12509.0 0.545650
\(808\) 43263.6 1.88367
\(809\) −26801.0 −1.16474 −0.582369 0.812925i \(-0.697874\pi\)
−0.582369 + 0.812925i \(0.697874\pi\)
\(810\) 7323.06 0.317662
\(811\) 19912.1 0.862157 0.431078 0.902315i \(-0.358133\pi\)
0.431078 + 0.902315i \(0.358133\pi\)
\(812\) −6412.12 −0.277120
\(813\) −15455.0 −0.666704
\(814\) 5082.06 0.218828
\(815\) 47856.7 2.05687
\(816\) 13283.1 0.569857
\(817\) −299.144 −0.0128100
\(818\) 23278.4 0.995001
\(819\) −9825.38 −0.419202
\(820\) 22150.2 0.943315
\(821\) 13978.8 0.594233 0.297116 0.954841i \(-0.403975\pi\)
0.297116 + 0.954841i \(0.403975\pi\)
\(822\) 6302.35 0.267420
\(823\) −45931.2 −1.94540 −0.972699 0.232070i \(-0.925450\pi\)
−0.972699 + 0.232070i \(0.925450\pi\)
\(824\) 23131.6 0.977946
\(825\) 7615.09 0.321362
\(826\) −27470.2 −1.15716
\(827\) 21005.1 0.883215 0.441607 0.897208i \(-0.354408\pi\)
0.441607 + 0.897208i \(0.354408\pi\)
\(828\) 2998.11 0.125835
\(829\) −8843.43 −0.370501 −0.185250 0.982691i \(-0.559310\pi\)
−0.185250 + 0.982691i \(0.559310\pi\)
\(830\) −57945.1 −2.42326
\(831\) −3100.24 −0.129418
\(832\) −39850.5 −1.66054
\(833\) −22.6416 −0.000941757 0
\(834\) −14225.8 −0.590647
\(835\) 7670.78 0.317914
\(836\) −1643.62 −0.0679972
\(837\) −1092.74 −0.0451264
\(838\) −21896.0 −0.902607
\(839\) −2322.54 −0.0955695 −0.0477848 0.998858i \(-0.515216\pi\)
−0.0477848 + 0.998858i \(0.515216\pi\)
\(840\) 35047.4 1.43958
\(841\) −23854.8 −0.978096
\(842\) −27350.3 −1.11942
\(843\) −20050.7 −0.819196
\(844\) 11574.5 0.472050
\(845\) −24060.4 −0.979529
\(846\) −7472.30 −0.303668
\(847\) −2241.63 −0.0909365
\(848\) −5367.42 −0.217356
\(849\) −23749.7 −0.960057
\(850\) 121066. 4.88535
\(851\) 2144.14 0.0863693
\(852\) 21979.5 0.883810
\(853\) −34381.4 −1.38007 −0.690033 0.723778i \(-0.742404\pi\)
−0.690033 + 0.723778i \(0.742404\pi\)
\(854\) −5416.72 −0.217045
\(855\) −1693.79 −0.0677502
\(856\) −32631.5 −1.30295
\(857\) −27698.0 −1.10402 −0.552010 0.833837i \(-0.686139\pi\)
−0.552010 + 0.833837i \(0.686139\pi\)
\(858\) −9321.19 −0.370886
\(859\) −34147.7 −1.35635 −0.678174 0.734901i \(-0.737229\pi\)
−0.678174 + 0.734901i \(0.737229\pi\)
\(860\) 8468.19 0.335771
\(861\) −4358.42 −0.172514
\(862\) −1495.77 −0.0591022
\(863\) 19295.6 0.761102 0.380551 0.924760i \(-0.375734\pi\)
0.380551 + 0.924760i \(0.375734\pi\)
\(864\) −1986.32 −0.0782129
\(865\) −63261.3 −2.48665
\(866\) −73649.2 −2.88996
\(867\) −21201.9 −0.830514
\(868\) −11228.0 −0.439059
\(869\) 3710.72 0.144853
\(870\) −6268.78 −0.244289
\(871\) 6090.54 0.236935
\(872\) 28853.5 1.12053
\(873\) 387.046 0.0150052
\(874\) −1063.90 −0.0411751
\(875\) 36955.5 1.42780
\(876\) −3584.00 −0.138233
\(877\) 44076.1 1.69709 0.848544 0.529125i \(-0.177480\pi\)
0.848544 + 0.529125i \(0.177480\pi\)
\(878\) 62754.9 2.41216
\(879\) 19127.2 0.733953
\(880\) 8392.99 0.321509
\(881\) 29485.7 1.12758 0.563791 0.825917i \(-0.309342\pi\)
0.563791 + 0.825917i \(0.309342\pi\)
\(882\) −8.92367 −0.000340675 0
\(883\) −24782.6 −0.944510 −0.472255 0.881462i \(-0.656560\pi\)
−0.472255 + 0.881462i \(0.656560\pi\)
\(884\) −96590.1 −3.67497
\(885\) −17504.7 −0.664876
\(886\) −37384.4 −1.41755
\(887\) 6772.25 0.256358 0.128179 0.991751i \(-0.459087\pi\)
0.128179 + 0.991751i \(0.459087\pi\)
\(888\) 9667.58 0.365341
\(889\) −10416.8 −0.392992
\(890\) 19834.3 0.747019
\(891\) −891.000 −0.0335013
\(892\) 79465.2 2.98284
\(893\) 1728.31 0.0647656
\(894\) −35724.7 −1.33648
\(895\) 88587.9 3.30857
\(896\) −49146.6 −1.83245
\(897\) −3932.65 −0.146385
\(898\) −79727.5 −2.96274
\(899\) 935.426 0.0347032
\(900\) 31100.9 1.15189
\(901\) 14522.9 0.536991
\(902\) −4134.76 −0.152630
\(903\) −1666.26 −0.0614059
\(904\) 35369.3 1.30129
\(905\) −25193.1 −0.925355
\(906\) −36567.9 −1.34093
\(907\) −25557.1 −0.935624 −0.467812 0.883828i \(-0.654957\pi\)
−0.467812 + 0.883828i \(0.654957\pi\)
\(908\) −24340.6 −0.889616
\(909\) −11646.3 −0.424953
\(910\) −98699.4 −3.59544
\(911\) −25703.4 −0.934788 −0.467394 0.884049i \(-0.654807\pi\)
−0.467394 + 0.884049i \(0.654807\pi\)
\(912\) −1210.89 −0.0439656
\(913\) 7050.20 0.255562
\(914\) 28153.1 1.01884
\(915\) −3451.67 −0.124709
\(916\) −41640.4 −1.50201
\(917\) −966.741 −0.0348142
\(918\) −14165.3 −0.509287
\(919\) −46893.9 −1.68323 −0.841614 0.540079i \(-0.818394\pi\)
−0.841614 + 0.540079i \(0.818394\pi\)
\(920\) 14027.9 0.502702
\(921\) 14794.5 0.529310
\(922\) 18445.5 0.658860
\(923\) −28830.8 −1.02814
\(924\) −9155.06 −0.325952
\(925\) 22242.3 0.790618
\(926\) 57522.3 2.04136
\(927\) −6226.88 −0.220623
\(928\) 1700.36 0.0601475
\(929\) −38581.3 −1.36255 −0.681277 0.732026i \(-0.738575\pi\)
−0.681277 + 0.732026i \(0.738575\pi\)
\(930\) −10977.0 −0.387043
\(931\) 2.06400 7.26585e−5 0
\(932\) −65397.1 −2.29845
\(933\) −14593.7 −0.512085
\(934\) 28318.7 0.992093
\(935\) −22709.4 −0.794307
\(936\) −17731.7 −0.619207
\(937\) 43555.9 1.51858 0.759290 0.650752i \(-0.225546\pi\)
0.759290 + 0.650752i \(0.225546\pi\)
\(938\) 9177.69 0.319469
\(939\) −18658.6 −0.648456
\(940\) −48925.0 −1.69762
\(941\) 13064.7 0.452599 0.226300 0.974058i \(-0.427337\pi\)
0.226300 + 0.974058i \(0.427337\pi\)
\(942\) −1956.32 −0.0676648
\(943\) −1744.48 −0.0602417
\(944\) −12514.1 −0.431462
\(945\) −9434.53 −0.324768
\(946\) −1580.75 −0.0543284
\(947\) 26424.6 0.906741 0.453370 0.891322i \(-0.350222\pi\)
0.453370 + 0.891322i \(0.350222\pi\)
\(948\) 15155.0 0.519210
\(949\) 4701.17 0.160808
\(950\) −11036.4 −0.376914
\(951\) −21283.8 −0.725735
\(952\) −67793.8 −2.30799
\(953\) 11812.2 0.401505 0.200752 0.979642i \(-0.435661\pi\)
0.200752 + 0.979642i \(0.435661\pi\)
\(954\) 5723.89 0.194253
\(955\) 19646.8 0.665713
\(956\) 21283.7 0.720045
\(957\) 762.726 0.0257632
\(958\) 86770.2 2.92632
\(959\) −8119.52 −0.273403
\(960\) −38265.3 −1.28646
\(961\) −28153.0 −0.945017
\(962\) −27225.5 −0.912459
\(963\) 8784.20 0.293943
\(964\) 13640.6 0.455740
\(965\) 47395.5 1.58105
\(966\) −5926.02 −0.197377
\(967\) −20141.3 −0.669805 −0.334903 0.942253i \(-0.608703\pi\)
−0.334903 + 0.942253i \(0.608703\pi\)
\(968\) −4045.42 −0.134323
\(969\) 3276.37 0.108620
\(970\) 3888.01 0.128697
\(971\) −29399.5 −0.971652 −0.485826 0.874055i \(-0.661481\pi\)
−0.485826 + 0.874055i \(0.661481\pi\)
\(972\) −3638.95 −0.120082
\(973\) 18327.6 0.603860
\(974\) 37058.3 1.21912
\(975\) −40795.4 −1.34000
\(976\) −2467.60 −0.0809283
\(977\) 23697.1 0.775986 0.387993 0.921662i \(-0.373168\pi\)
0.387993 + 0.921662i \(0.373168\pi\)
\(978\) −36485.0 −1.19290
\(979\) −2413.25 −0.0787822
\(980\) −58.4279 −0.00190450
\(981\) −7767.17 −0.252790
\(982\) 53513.8 1.73900
\(983\) 18025.7 0.584872 0.292436 0.956285i \(-0.405534\pi\)
0.292436 + 0.956285i \(0.405534\pi\)
\(984\) −7865.55 −0.254822
\(985\) 64669.5 2.09192
\(986\) 12126.0 0.391653
\(987\) 9626.81 0.310461
\(988\) 8805.15 0.283531
\(989\) −666.927 −0.0214429
\(990\) −8950.40 −0.287336
\(991\) 5295.79 0.169754 0.0848771 0.996391i \(-0.472950\pi\)
0.0848771 + 0.996391i \(0.472950\pi\)
\(992\) 2977.42 0.0952956
\(993\) −18.3810 −0.000587414 0
\(994\) −43444.4 −1.38629
\(995\) 66531.8 2.11980
\(996\) 28793.8 0.916032
\(997\) −55027.5 −1.74798 −0.873990 0.485943i \(-0.838476\pi\)
−0.873990 + 0.485943i \(0.838476\pi\)
\(998\) −23601.6 −0.748591
\(999\) −2602.45 −0.0824202
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 2013.4.a.e.1.4 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.4.a.e.1.4 38 1.1 even 1 trivial