Properties

Label 177.4.a.b.1.6
Level $177$
Weight $4$
Character 177.1
Self dual yes
Analytic conductor $10.443$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4433380710\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 41 x^{5} - 7 x^{4} + 484 x^{3} + 63 x^{2} - 1736 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-3.39497\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.39497 q^{2} -3.00000 q^{3} +3.52580 q^{4} -6.09684 q^{5} -10.1849 q^{6} +1.40309 q^{7} -15.1898 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.39497 q^{2} -3.00000 q^{3} +3.52580 q^{4} -6.09684 q^{5} -10.1849 q^{6} +1.40309 q^{7} -15.1898 q^{8} +9.00000 q^{9} -20.6986 q^{10} +1.38217 q^{11} -10.5774 q^{12} -63.1312 q^{13} +4.76343 q^{14} +18.2905 q^{15} -79.7751 q^{16} -121.163 q^{17} +30.5547 q^{18} -44.5277 q^{19} -21.4963 q^{20} -4.20926 q^{21} +4.69240 q^{22} +138.130 q^{23} +45.5693 q^{24} -87.8285 q^{25} -214.328 q^{26} -27.0000 q^{27} +4.94701 q^{28} +6.31405 q^{29} +62.0957 q^{30} +246.448 q^{31} -149.316 q^{32} -4.14650 q^{33} -411.345 q^{34} -8.55440 q^{35} +31.7322 q^{36} +300.643 q^{37} -151.170 q^{38} +189.394 q^{39} +92.6096 q^{40} +110.943 q^{41} -14.2903 q^{42} -156.551 q^{43} +4.87324 q^{44} -54.8716 q^{45} +468.946 q^{46} -338.548 q^{47} +239.325 q^{48} -341.031 q^{49} -298.175 q^{50} +363.489 q^{51} -222.588 q^{52} +484.870 q^{53} -91.6641 q^{54} -8.42684 q^{55} -21.3126 q^{56} +133.583 q^{57} +21.4360 q^{58} +59.0000 q^{59} +64.4888 q^{60} +606.636 q^{61} +836.682 q^{62} +12.6278 q^{63} +131.279 q^{64} +384.901 q^{65} -14.0772 q^{66} -724.593 q^{67} -427.197 q^{68} -414.389 q^{69} -29.0419 q^{70} -345.720 q^{71} -136.708 q^{72} -1158.89 q^{73} +1020.67 q^{74} +263.486 q^{75} -156.996 q^{76} +1.93930 q^{77} +642.985 q^{78} -145.495 q^{79} +486.376 q^{80} +81.0000 q^{81} +376.649 q^{82} -858.838 q^{83} -14.8410 q^{84} +738.712 q^{85} -531.486 q^{86} -18.9422 q^{87} -20.9948 q^{88} +6.88345 q^{89} -186.287 q^{90} -88.5786 q^{91} +487.018 q^{92} -739.343 q^{93} -1149.36 q^{94} +271.479 q^{95} +447.948 q^{96} -540.926 q^{97} -1157.79 q^{98} +12.4395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} - 71q^{10} - 5q^{11} - 78q^{12} - 67q^{13} - 65q^{14} + 6q^{15} - 94q^{16} - 23q^{17} - 176q^{19} - 207q^{20} + 177q^{21} - 704q^{22} - 218q^{23} + 63q^{24} - 183q^{25} + 58q^{26} - 189q^{27} - 938q^{28} + 168q^{29} + 213q^{30} - 604q^{31} - 448q^{32} + 15q^{33} - 610q^{34} - 336q^{35} + 234q^{36} - 505q^{37} - 453q^{38} + 201q^{39} - 1080q^{40} - 265q^{41} + 195q^{42} - 493q^{43} + 504q^{44} - 18q^{45} + 381q^{46} - 244q^{47} + 282q^{48} + 770q^{49} + 1639q^{50} + 69q^{51} + 160q^{52} + 686q^{53} - 116q^{55} + 2190q^{56} + 528q^{57} + 1584q^{58} + 413q^{59} + 621q^{60} - 838q^{61} + 286q^{62} - 531q^{63} + 205q^{64} + 490q^{65} + 2112q^{66} - 1504q^{67} + 3047q^{68} + 654q^{69} + 1530q^{70} - 1267q^{71} - 189q^{72} - 666q^{73} + 528q^{74} + 549q^{75} - 64q^{76} + 1109q^{77} - 174q^{78} - 2741q^{79} + 1213q^{80} + 567q^{81} + 953q^{82} - 2025q^{83} + 2814q^{84} - 1274q^{85} + 4394q^{86} - 504q^{87} - 1639q^{88} + 616q^{89} - 639q^{90} - 2415q^{91} + 218q^{92} + 1812q^{93} + 900q^{94} + 2554q^{95} + 1344q^{96} - 1298q^{97} - 172q^{98} - 45q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.39497 1.20030 0.600151 0.799887i \(-0.295107\pi\)
0.600151 + 0.799887i \(0.295107\pi\)
\(3\) −3.00000 −0.577350
\(4\) 3.52580 0.440725
\(5\) −6.09684 −0.545318 −0.272659 0.962111i \(-0.587903\pi\)
−0.272659 + 0.962111i \(0.587903\pi\)
\(6\) −10.1849 −0.692995
\(7\) 1.40309 0.0757596 0.0378798 0.999282i \(-0.487940\pi\)
0.0378798 + 0.999282i \(0.487940\pi\)
\(8\) −15.1898 −0.671299
\(9\) 9.00000 0.333333
\(10\) −20.6986 −0.654547
\(11\) 1.38217 0.0378853 0.0189427 0.999821i \(-0.493970\pi\)
0.0189427 + 0.999821i \(0.493970\pi\)
\(12\) −10.5774 −0.254453
\(13\) −63.1312 −1.34688 −0.673441 0.739241i \(-0.735184\pi\)
−0.673441 + 0.739241i \(0.735184\pi\)
\(14\) 4.76343 0.0909344
\(15\) 18.2905 0.314840
\(16\) −79.7751 −1.24649
\(17\) −121.163 −1.72861 −0.864305 0.502969i \(-0.832241\pi\)
−0.864305 + 0.502969i \(0.832241\pi\)
\(18\) 30.5547 0.400101
\(19\) −44.5277 −0.537651 −0.268825 0.963189i \(-0.586635\pi\)
−0.268825 + 0.963189i \(0.586635\pi\)
\(20\) −21.4963 −0.240335
\(21\) −4.20926 −0.0437398
\(22\) 4.69240 0.0454738
\(23\) 138.130 1.25226 0.626131 0.779717i \(-0.284637\pi\)
0.626131 + 0.779717i \(0.284637\pi\)
\(24\) 45.5693 0.387575
\(25\) −87.8285 −0.702628
\(26\) −214.328 −1.61666
\(27\) −27.0000 −0.192450
\(28\) 4.94701 0.0333892
\(29\) 6.31405 0.0404307 0.0202153 0.999796i \(-0.493565\pi\)
0.0202153 + 0.999796i \(0.493565\pi\)
\(30\) 62.0957 0.377903
\(31\) 246.448 1.42785 0.713924 0.700223i \(-0.246916\pi\)
0.713924 + 0.700223i \(0.246916\pi\)
\(32\) −149.316 −0.824862
\(33\) −4.14650 −0.0218731
\(34\) −411.345 −2.07485
\(35\) −8.55440 −0.0413131
\(36\) 31.7322 0.146908
\(37\) 300.643 1.33582 0.667910 0.744242i \(-0.267189\pi\)
0.667910 + 0.744242i \(0.267189\pi\)
\(38\) −151.170 −0.645343
\(39\) 189.394 0.777622
\(40\) 92.6096 0.366071
\(41\) 110.943 0.422596 0.211298 0.977422i \(-0.432231\pi\)
0.211298 + 0.977422i \(0.432231\pi\)
\(42\) −14.2903 −0.0525010
\(43\) −156.551 −0.555205 −0.277602 0.960696i \(-0.589540\pi\)
−0.277602 + 0.960696i \(0.589540\pi\)
\(44\) 4.87324 0.0166970
\(45\) −54.8716 −0.181773
\(46\) 468.946 1.50309
\(47\) −338.548 −1.05069 −0.525344 0.850890i \(-0.676063\pi\)
−0.525344 + 0.850890i \(0.676063\pi\)
\(48\) 239.325 0.719659
\(49\) −341.031 −0.994260
\(50\) −298.175 −0.843366
\(51\) 363.489 0.998013
\(52\) −222.588 −0.593604
\(53\) 484.870 1.25664 0.628322 0.777954i \(-0.283742\pi\)
0.628322 + 0.777954i \(0.283742\pi\)
\(54\) −91.6641 −0.230998
\(55\) −8.42684 −0.0206595
\(56\) −21.3126 −0.0508573
\(57\) 133.583 0.310413
\(58\) 21.4360 0.0485290
\(59\) 59.0000 0.130189
\(60\) 64.4888 0.138758
\(61\) 606.636 1.27331 0.636654 0.771150i \(-0.280318\pi\)
0.636654 + 0.771150i \(0.280318\pi\)
\(62\) 836.682 1.71385
\(63\) 12.6278 0.0252532
\(64\) 131.279 0.256403
\(65\) 384.901 0.734479
\(66\) −14.0772 −0.0262543
\(67\) −724.593 −1.32124 −0.660620 0.750720i \(-0.729707\pi\)
−0.660620 + 0.750720i \(0.729707\pi\)
\(68\) −427.197 −0.761842
\(69\) −414.389 −0.722994
\(70\) −29.0419 −0.0495882
\(71\) −345.720 −0.577879 −0.288940 0.957347i \(-0.593303\pi\)
−0.288940 + 0.957347i \(0.593303\pi\)
\(72\) −136.708 −0.223766
\(73\) −1158.89 −1.85806 −0.929029 0.370008i \(-0.879355\pi\)
−0.929029 + 0.370008i \(0.879355\pi\)
\(74\) 1020.67 1.60339
\(75\) 263.486 0.405663
\(76\) −156.996 −0.236956
\(77\) 1.93930 0.00287018
\(78\) 642.985 0.933382
\(79\) −145.495 −0.207209 −0.103605 0.994619i \(-0.533038\pi\)
−0.103605 + 0.994619i \(0.533038\pi\)
\(80\) 486.376 0.679732
\(81\) 81.0000 0.111111
\(82\) 376.649 0.507243
\(83\) −858.838 −1.13578 −0.567890 0.823104i \(-0.692240\pi\)
−0.567890 + 0.823104i \(0.692240\pi\)
\(84\) −14.8410 −0.0192772
\(85\) 738.712 0.942642
\(86\) −531.486 −0.666413
\(87\) −18.9422 −0.0233427
\(88\) −20.9948 −0.0254324
\(89\) 6.88345 0.00819825 0.00409912 0.999992i \(-0.498695\pi\)
0.00409912 + 0.999992i \(0.498695\pi\)
\(90\) −186.287 −0.218182
\(91\) −88.5786 −0.102039
\(92\) 487.018 0.551904
\(93\) −739.343 −0.824369
\(94\) −1149.36 −1.26114
\(95\) 271.479 0.293191
\(96\) 447.948 0.476234
\(97\) −540.926 −0.566214 −0.283107 0.959088i \(-0.591365\pi\)
−0.283107 + 0.959088i \(0.591365\pi\)
\(98\) −1157.79 −1.19341
\(99\) 12.4395 0.0126284
\(100\) −309.666 −0.309666
\(101\) 1398.33 1.37762 0.688808 0.724944i \(-0.258135\pi\)
0.688808 + 0.724944i \(0.258135\pi\)
\(102\) 1234.03 1.19792
\(103\) −1320.13 −1.26287 −0.631436 0.775428i \(-0.717534\pi\)
−0.631436 + 0.775428i \(0.717534\pi\)
\(104\) 958.948 0.904160
\(105\) 25.6632 0.0238521
\(106\) 1646.12 1.50835
\(107\) 1276.58 1.15338 0.576690 0.816963i \(-0.304344\pi\)
0.576690 + 0.816963i \(0.304344\pi\)
\(108\) −95.1966 −0.0848176
\(109\) −770.303 −0.676896 −0.338448 0.940985i \(-0.609902\pi\)
−0.338448 + 0.940985i \(0.609902\pi\)
\(110\) −28.6089 −0.0247977
\(111\) −901.928 −0.771236
\(112\) −111.931 −0.0944333
\(113\) 892.165 0.742724 0.371362 0.928488i \(-0.378891\pi\)
0.371362 + 0.928488i \(0.378891\pi\)
\(114\) 453.511 0.372589
\(115\) −842.155 −0.682882
\(116\) 22.2621 0.0178188
\(117\) −568.181 −0.448960
\(118\) 200.303 0.156266
\(119\) −170.002 −0.130959
\(120\) −277.829 −0.211351
\(121\) −1329.09 −0.998565
\(122\) 2059.51 1.52835
\(123\) −332.830 −0.243986
\(124\) 868.925 0.629289
\(125\) 1297.58 0.928474
\(126\) 42.8709 0.0303115
\(127\) 699.506 0.488749 0.244374 0.969681i \(-0.421417\pi\)
0.244374 + 0.969681i \(0.421417\pi\)
\(128\) 1640.21 1.13262
\(129\) 469.653 0.320548
\(130\) 1306.73 0.881596
\(131\) 2441.72 1.62850 0.814252 0.580512i \(-0.197147\pi\)
0.814252 + 0.580512i \(0.197147\pi\)
\(132\) −14.6197 −0.00964002
\(133\) −62.4763 −0.0407322
\(134\) −2459.97 −1.58589
\(135\) 164.615 0.104947
\(136\) 1840.44 1.16041
\(137\) −410.102 −0.255747 −0.127874 0.991790i \(-0.540815\pi\)
−0.127874 + 0.991790i \(0.540815\pi\)
\(138\) −1406.84 −0.867812
\(139\) −2421.14 −1.47740 −0.738699 0.674035i \(-0.764560\pi\)
−0.738699 + 0.674035i \(0.764560\pi\)
\(140\) −30.1611 −0.0182077
\(141\) 1015.64 0.606615
\(142\) −1173.71 −0.693630
\(143\) −87.2578 −0.0510270
\(144\) −717.976 −0.415495
\(145\) −38.4958 −0.0220476
\(146\) −3934.40 −2.23023
\(147\) 1023.09 0.574037
\(148\) 1060.01 0.588729
\(149\) −916.292 −0.503796 −0.251898 0.967754i \(-0.581055\pi\)
−0.251898 + 0.967754i \(0.581055\pi\)
\(150\) 894.525 0.486918
\(151\) −2476.55 −1.33470 −0.667348 0.744746i \(-0.732571\pi\)
−0.667348 + 0.744746i \(0.732571\pi\)
\(152\) 676.366 0.360924
\(153\) −1090.47 −0.576203
\(154\) 6.58385 0.00344508
\(155\) −1502.55 −0.778632
\(156\) 667.765 0.342718
\(157\) −1892.25 −0.961896 −0.480948 0.876749i \(-0.659707\pi\)
−0.480948 + 0.876749i \(0.659707\pi\)
\(158\) −493.952 −0.248713
\(159\) −1454.61 −0.725523
\(160\) 910.356 0.449812
\(161\) 193.808 0.0948709
\(162\) 274.992 0.133367
\(163\) 1260.29 0.605605 0.302802 0.953053i \(-0.402078\pi\)
0.302802 + 0.953053i \(0.402078\pi\)
\(164\) 391.164 0.186249
\(165\) 25.2805 0.0119278
\(166\) −2915.73 −1.36328
\(167\) −2638.25 −1.22248 −0.611239 0.791446i \(-0.709329\pi\)
−0.611239 + 0.791446i \(0.709329\pi\)
\(168\) 63.9377 0.0293625
\(169\) 1788.55 0.814089
\(170\) 2507.90 1.13146
\(171\) −400.750 −0.179217
\(172\) −551.968 −0.244693
\(173\) −2057.01 −0.903996 −0.451998 0.892019i \(-0.649289\pi\)
−0.451998 + 0.892019i \(0.649289\pi\)
\(174\) −64.3080 −0.0280182
\(175\) −123.231 −0.0532308
\(176\) −110.262 −0.0472235
\(177\) −177.000 −0.0751646
\(178\) 23.3691 0.00984038
\(179\) −2195.47 −0.916745 −0.458372 0.888760i \(-0.651567\pi\)
−0.458372 + 0.888760i \(0.651567\pi\)
\(180\) −193.466 −0.0801118
\(181\) −1103.48 −0.453153 −0.226577 0.973993i \(-0.572753\pi\)
−0.226577 + 0.973993i \(0.572753\pi\)
\(182\) −300.722 −0.122478
\(183\) −1819.91 −0.735144
\(184\) −2098.16 −0.840643
\(185\) −1832.97 −0.728447
\(186\) −2510.04 −0.989491
\(187\) −167.467 −0.0654889
\(188\) −1193.65 −0.463064
\(189\) −37.8834 −0.0145799
\(190\) 921.661 0.351917
\(191\) 780.850 0.295813 0.147907 0.989001i \(-0.452747\pi\)
0.147907 + 0.989001i \(0.452747\pi\)
\(192\) −393.836 −0.148035
\(193\) 3823.85 1.42615 0.713074 0.701089i \(-0.247302\pi\)
0.713074 + 0.701089i \(0.247302\pi\)
\(194\) −1836.43 −0.679627
\(195\) −1154.70 −0.424052
\(196\) −1202.41 −0.438196
\(197\) −3145.99 −1.13778 −0.568889 0.822414i \(-0.692627\pi\)
−0.568889 + 0.822414i \(0.692627\pi\)
\(198\) 42.2316 0.0151579
\(199\) 246.190 0.0876981 0.0438491 0.999038i \(-0.486038\pi\)
0.0438491 + 0.999038i \(0.486038\pi\)
\(200\) 1334.09 0.471673
\(201\) 2173.78 0.762819
\(202\) 4747.29 1.65355
\(203\) 8.85916 0.00306301
\(204\) 1281.59 0.439850
\(205\) −676.404 −0.230449
\(206\) −4481.78 −1.51583
\(207\) 1243.17 0.417421
\(208\) 5036.30 1.67887
\(209\) −61.5447 −0.0203691
\(210\) 87.1257 0.0286297
\(211\) 376.324 0.122783 0.0613914 0.998114i \(-0.480446\pi\)
0.0613914 + 0.998114i \(0.480446\pi\)
\(212\) 1709.56 0.553834
\(213\) 1037.16 0.333639
\(214\) 4333.94 1.38440
\(215\) 954.467 0.302763
\(216\) 410.123 0.129192
\(217\) 345.787 0.108173
\(218\) −2615.15 −0.812479
\(219\) 3476.68 1.07275
\(220\) −29.7114 −0.00910518
\(221\) 7649.17 2.32823
\(222\) −3062.01 −0.925716
\(223\) −2989.63 −0.897758 −0.448879 0.893592i \(-0.648177\pi\)
−0.448879 + 0.893592i \(0.648177\pi\)
\(224\) −209.503 −0.0624912
\(225\) −790.457 −0.234209
\(226\) 3028.87 0.891494
\(227\) 3596.10 1.05146 0.525730 0.850651i \(-0.323792\pi\)
0.525730 + 0.850651i \(0.323792\pi\)
\(228\) 470.988 0.136807
\(229\) −1227.07 −0.354092 −0.177046 0.984203i \(-0.556654\pi\)
−0.177046 + 0.984203i \(0.556654\pi\)
\(230\) −2859.09 −0.819664
\(231\) −5.81789 −0.00165710
\(232\) −95.9089 −0.0271411
\(233\) −402.265 −0.113104 −0.0565521 0.998400i \(-0.518011\pi\)
−0.0565521 + 0.998400i \(0.518011\pi\)
\(234\) −1928.96 −0.538888
\(235\) 2064.07 0.572959
\(236\) 208.022 0.0573775
\(237\) 436.486 0.119632
\(238\) −577.152 −0.157190
\(239\) −4244.60 −1.14879 −0.574393 0.818579i \(-0.694762\pi\)
−0.574393 + 0.818579i \(0.694762\pi\)
\(240\) −1459.13 −0.392443
\(241\) 4745.99 1.26853 0.634266 0.773115i \(-0.281302\pi\)
0.634266 + 0.773115i \(0.281302\pi\)
\(242\) −4512.22 −1.19858
\(243\) −243.000 −0.0641500
\(244\) 2138.88 0.561179
\(245\) 2079.21 0.542188
\(246\) −1129.95 −0.292857
\(247\) 2811.09 0.724152
\(248\) −3743.48 −0.958513
\(249\) 2576.52 0.655743
\(250\) 4405.25 1.11445
\(251\) 2511.21 0.631499 0.315749 0.948843i \(-0.397744\pi\)
0.315749 + 0.948843i \(0.397744\pi\)
\(252\) 44.5231 0.0111297
\(253\) 190.918 0.0474424
\(254\) 2374.80 0.586646
\(255\) −2216.14 −0.544235
\(256\) 4518.24 1.10309
\(257\) 6329.37 1.53625 0.768123 0.640303i \(-0.221191\pi\)
0.768123 + 0.640303i \(0.221191\pi\)
\(258\) 1594.46 0.384754
\(259\) 421.828 0.101201
\(260\) 1357.09 0.323703
\(261\) 56.8265 0.0134769
\(262\) 8289.56 1.95470
\(263\) −1919.91 −0.450140 −0.225070 0.974343i \(-0.572261\pi\)
−0.225070 + 0.974343i \(0.572261\pi\)
\(264\) 62.9843 0.0146834
\(265\) −2956.18 −0.685270
\(266\) −212.105 −0.0488909
\(267\) −20.6503 −0.00473326
\(268\) −2554.77 −0.582304
\(269\) 4709.65 1.06748 0.533741 0.845648i \(-0.320786\pi\)
0.533741 + 0.845648i \(0.320786\pi\)
\(270\) 558.862 0.125968
\(271\) −4913.41 −1.10136 −0.550680 0.834717i \(-0.685631\pi\)
−0.550680 + 0.834717i \(0.685631\pi\)
\(272\) 9665.80 2.15469
\(273\) 265.736 0.0589123
\(274\) −1392.28 −0.306974
\(275\) −121.393 −0.0266193
\(276\) −1461.05 −0.318642
\(277\) 2089.05 0.453136 0.226568 0.973995i \(-0.427250\pi\)
0.226568 + 0.973995i \(0.427250\pi\)
\(278\) −8219.69 −1.77332
\(279\) 2218.03 0.475949
\(280\) 129.939 0.0277334
\(281\) −372.231 −0.0790229 −0.0395114 0.999219i \(-0.512580\pi\)
−0.0395114 + 0.999219i \(0.512580\pi\)
\(282\) 3448.08 0.728121
\(283\) −5381.33 −1.13034 −0.565171 0.824974i \(-0.691190\pi\)
−0.565171 + 0.824974i \(0.691190\pi\)
\(284\) −1218.94 −0.254686
\(285\) −814.436 −0.169274
\(286\) −296.237 −0.0612478
\(287\) 155.663 0.0320157
\(288\) −1343.84 −0.274954
\(289\) 9767.49 1.98809
\(290\) −130.692 −0.0264638
\(291\) 1622.78 0.326904
\(292\) −4086.03 −0.818893
\(293\) 7468.20 1.48907 0.744534 0.667584i \(-0.232672\pi\)
0.744534 + 0.667584i \(0.232672\pi\)
\(294\) 3473.37 0.689017
\(295\) −359.714 −0.0709944
\(296\) −4566.69 −0.896734
\(297\) −37.3185 −0.00729103
\(298\) −3110.78 −0.604707
\(299\) −8720.30 −1.68665
\(300\) 928.998 0.178786
\(301\) −219.655 −0.0420621
\(302\) −8407.82 −1.60204
\(303\) −4194.99 −0.795366
\(304\) 3552.21 0.670174
\(305\) −3698.56 −0.694358
\(306\) −3702.10 −0.691618
\(307\) 4475.24 0.831972 0.415986 0.909371i \(-0.363437\pi\)
0.415986 + 0.909371i \(0.363437\pi\)
\(308\) 6.83758 0.00126496
\(309\) 3960.38 0.729120
\(310\) −5101.12 −0.934593
\(311\) −2483.35 −0.452791 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(312\) −2876.84 −0.522017
\(313\) −4675.53 −0.844334 −0.422167 0.906518i \(-0.638730\pi\)
−0.422167 + 0.906518i \(0.638730\pi\)
\(314\) −6424.11 −1.15457
\(315\) −76.9896 −0.0137710
\(316\) −512.988 −0.0913222
\(317\) 5009.94 0.887655 0.443827 0.896112i \(-0.353620\pi\)
0.443827 + 0.896112i \(0.353620\pi\)
\(318\) −4938.36 −0.870847
\(319\) 8.72706 0.00153173
\(320\) −800.384 −0.139821
\(321\) −3829.74 −0.665904
\(322\) 657.972 0.113874
\(323\) 5395.12 0.929388
\(324\) 285.590 0.0489695
\(325\) 5544.72 0.946356
\(326\) 4278.65 0.726909
\(327\) 2310.91 0.390806
\(328\) −1685.20 −0.283688
\(329\) −475.012 −0.0795996
\(330\) 85.8266 0.0143170
\(331\) 6071.68 1.00825 0.504124 0.863631i \(-0.331816\pi\)
0.504124 + 0.863631i \(0.331816\pi\)
\(332\) −3028.09 −0.500567
\(333\) 2705.78 0.445273
\(334\) −8956.77 −1.46734
\(335\) 4417.73 0.720497
\(336\) 335.794 0.0545211
\(337\) −325.012 −0.0525358 −0.0262679 0.999655i \(-0.508362\pi\)
−0.0262679 + 0.999655i \(0.508362\pi\)
\(338\) 6072.08 0.977152
\(339\) −2676.50 −0.428812
\(340\) 2604.55 0.415446
\(341\) 340.631 0.0540945
\(342\) −1360.53 −0.215114
\(343\) −959.756 −0.151084
\(344\) 2377.97 0.372708
\(345\) 2526.47 0.394262
\(346\) −6983.47 −1.08507
\(347\) −8653.52 −1.33875 −0.669374 0.742926i \(-0.733438\pi\)
−0.669374 + 0.742926i \(0.733438\pi\)
\(348\) −66.7863 −0.0102877
\(349\) −6206.30 −0.951907 −0.475954 0.879470i \(-0.657897\pi\)
−0.475954 + 0.879470i \(0.657897\pi\)
\(350\) −418.365 −0.0638931
\(351\) 1704.54 0.259207
\(352\) −206.379 −0.0312501
\(353\) −4047.05 −0.610206 −0.305103 0.952319i \(-0.598691\pi\)
−0.305103 + 0.952319i \(0.598691\pi\)
\(354\) −600.909 −0.0902202
\(355\) 2107.80 0.315128
\(356\) 24.2697 0.00361317
\(357\) 510.007 0.0756091
\(358\) −7453.56 −1.10037
\(359\) 3831.18 0.563237 0.281619 0.959526i \(-0.409129\pi\)
0.281619 + 0.959526i \(0.409129\pi\)
\(360\) 833.486 0.122024
\(361\) −4876.28 −0.710932
\(362\) −3746.27 −0.543921
\(363\) 3987.27 0.576522
\(364\) −312.311 −0.0449712
\(365\) 7065.59 1.01323
\(366\) −6178.53 −0.882395
\(367\) 3771.70 0.536461 0.268231 0.963355i \(-0.413561\pi\)
0.268231 + 0.963355i \(0.413561\pi\)
\(368\) −11019.3 −1.56093
\(369\) 998.489 0.140865
\(370\) −6222.87 −0.874356
\(371\) 680.315 0.0952027
\(372\) −2606.78 −0.363320
\(373\) −5563.11 −0.772244 −0.386122 0.922448i \(-0.626186\pi\)
−0.386122 + 0.922448i \(0.626186\pi\)
\(374\) −568.546 −0.0786065
\(375\) −3892.75 −0.536055
\(376\) 5142.46 0.705325
\(377\) −398.614 −0.0544553
\(378\) −128.613 −0.0175003
\(379\) 12774.1 1.73129 0.865646 0.500656i \(-0.166908\pi\)
0.865646 + 0.500656i \(0.166908\pi\)
\(380\) 957.180 0.129217
\(381\) −2098.52 −0.282179
\(382\) 2650.96 0.355065
\(383\) −10186.2 −1.35899 −0.679494 0.733681i \(-0.737801\pi\)
−0.679494 + 0.733681i \(0.737801\pi\)
\(384\) −4920.64 −0.653920
\(385\) −11.8236 −0.00156516
\(386\) 12981.8 1.71181
\(387\) −1408.96 −0.185068
\(388\) −1907.20 −0.249545
\(389\) 10360.7 1.35040 0.675202 0.737633i \(-0.264056\pi\)
0.675202 + 0.737633i \(0.264056\pi\)
\(390\) −3920.18 −0.508990
\(391\) −16736.2 −2.16467
\(392\) 5180.18 0.667446
\(393\) −7325.16 −0.940217
\(394\) −10680.5 −1.36568
\(395\) 887.063 0.112995
\(396\) 43.8592 0.00556567
\(397\) −1360.01 −0.171932 −0.0859658 0.996298i \(-0.527398\pi\)
−0.0859658 + 0.996298i \(0.527398\pi\)
\(398\) 835.806 0.105264
\(399\) 187.429 0.0235167
\(400\) 7006.53 0.875816
\(401\) 2226.21 0.277236 0.138618 0.990346i \(-0.455734\pi\)
0.138618 + 0.990346i \(0.455734\pi\)
\(402\) 7379.91 0.915613
\(403\) −15558.5 −1.92314
\(404\) 4930.24 0.607150
\(405\) −493.844 −0.0605909
\(406\) 30.0766 0.00367654
\(407\) 415.538 0.0506079
\(408\) −5521.31 −0.669965
\(409\) −3590.70 −0.434104 −0.217052 0.976160i \(-0.569644\pi\)
−0.217052 + 0.976160i \(0.569644\pi\)
\(410\) −2296.37 −0.276609
\(411\) 1230.31 0.147656
\(412\) −4654.50 −0.556580
\(413\) 82.7821 0.00986306
\(414\) 4220.51 0.501031
\(415\) 5236.20 0.619362
\(416\) 9426.50 1.11099
\(417\) 7263.42 0.852977
\(418\) −208.942 −0.0244490
\(419\) −8552.13 −0.997133 −0.498567 0.866851i \(-0.666140\pi\)
−0.498567 + 0.866851i \(0.666140\pi\)
\(420\) 90.4834 0.0105122
\(421\) −13370.6 −1.54785 −0.773924 0.633279i \(-0.781709\pi\)
−0.773924 + 0.633279i \(0.781709\pi\)
\(422\) 1277.61 0.147377
\(423\) −3046.93 −0.350229
\(424\) −7365.06 −0.843583
\(425\) 10641.6 1.21457
\(426\) 3521.12 0.400467
\(427\) 851.163 0.0964652
\(428\) 4500.97 0.508323
\(429\) 261.773 0.0294605
\(430\) 3240.38 0.363407
\(431\) 17203.4 1.92264 0.961321 0.275431i \(-0.0888205\pi\)
0.961321 + 0.275431i \(0.0888205\pi\)
\(432\) 2153.93 0.239886
\(433\) 685.066 0.0760327 0.0380164 0.999277i \(-0.487896\pi\)
0.0380164 + 0.999277i \(0.487896\pi\)
\(434\) 1173.94 0.129841
\(435\) 115.487 0.0127292
\(436\) −2715.93 −0.298325
\(437\) −6150.61 −0.673280
\(438\) 11803.2 1.28762
\(439\) −5680.31 −0.617555 −0.308777 0.951134i \(-0.599920\pi\)
−0.308777 + 0.951134i \(0.599920\pi\)
\(440\) 128.002 0.0138687
\(441\) −3069.28 −0.331420
\(442\) 25968.7 2.79458
\(443\) 11657.0 1.25021 0.625105 0.780541i \(-0.285056\pi\)
0.625105 + 0.780541i \(0.285056\pi\)
\(444\) −3180.02 −0.339903
\(445\) −41.9673 −0.00447065
\(446\) −10149.7 −1.07758
\(447\) 2748.88 0.290867
\(448\) 184.195 0.0194250
\(449\) 277.511 0.0291682 0.0145841 0.999894i \(-0.495358\pi\)
0.0145841 + 0.999894i \(0.495358\pi\)
\(450\) −2683.57 −0.281122
\(451\) 153.342 0.0160102
\(452\) 3145.60 0.327337
\(453\) 7429.66 0.770587
\(454\) 12208.6 1.26207
\(455\) 540.050 0.0556438
\(456\) −2029.10 −0.208380
\(457\) 6866.73 0.702871 0.351436 0.936212i \(-0.385694\pi\)
0.351436 + 0.936212i \(0.385694\pi\)
\(458\) −4165.86 −0.425017
\(459\) 3271.40 0.332671
\(460\) −2969.27 −0.300963
\(461\) −9048.95 −0.914211 −0.457106 0.889412i \(-0.651114\pi\)
−0.457106 + 0.889412i \(0.651114\pi\)
\(462\) −19.7516 −0.00198902
\(463\) 15273.3 1.53307 0.766535 0.642203i \(-0.221979\pi\)
0.766535 + 0.642203i \(0.221979\pi\)
\(464\) −503.704 −0.0503963
\(465\) 4507.66 0.449543
\(466\) −1365.68 −0.135759
\(467\) −15724.0 −1.55807 −0.779035 0.626981i \(-0.784290\pi\)
−0.779035 + 0.626981i \(0.784290\pi\)
\(468\) −2003.29 −0.197868
\(469\) −1016.67 −0.100097
\(470\) 7007.46 0.687724
\(471\) 5676.74 0.555351
\(472\) −896.196 −0.0873957
\(473\) −216.379 −0.0210341
\(474\) 1481.86 0.143595
\(475\) 3910.81 0.377769
\(476\) −599.394 −0.0577168
\(477\) 4363.83 0.418881
\(478\) −14410.3 −1.37889
\(479\) −60.2922 −0.00575119 −0.00287560 0.999996i \(-0.500915\pi\)
−0.00287560 + 0.999996i \(0.500915\pi\)
\(480\) −2731.07 −0.259699
\(481\) −18979.9 −1.79919
\(482\) 16112.5 1.52262
\(483\) −581.424 −0.0547738
\(484\) −4686.11 −0.440093
\(485\) 3297.94 0.308767
\(486\) −824.977 −0.0769994
\(487\) 7565.04 0.703911 0.351955 0.936017i \(-0.385517\pi\)
0.351955 + 0.936017i \(0.385517\pi\)
\(488\) −9214.65 −0.854770
\(489\) −3780.87 −0.349646
\(490\) 7058.86 0.650790
\(491\) 1670.92 0.153579 0.0767897 0.997047i \(-0.475533\pi\)
0.0767897 + 0.997047i \(0.475533\pi\)
\(492\) −1173.49 −0.107531
\(493\) −765.030 −0.0698889
\(494\) 9543.56 0.869201
\(495\) −75.8416 −0.00688652
\(496\) −19660.4 −1.77979
\(497\) −485.075 −0.0437799
\(498\) 8747.18 0.787090
\(499\) −4120.64 −0.369669 −0.184835 0.982770i \(-0.559175\pi\)
−0.184835 + 0.982770i \(0.559175\pi\)
\(500\) 4575.02 0.409202
\(501\) 7914.75 0.705798
\(502\) 8525.48 0.757989
\(503\) −13860.7 −1.22866 −0.614331 0.789049i \(-0.710574\pi\)
−0.614331 + 0.789049i \(0.710574\pi\)
\(504\) −191.813 −0.0169524
\(505\) −8525.40 −0.751239
\(506\) 648.161 0.0569452
\(507\) −5365.66 −0.470014
\(508\) 2466.32 0.215404
\(509\) 17988.0 1.56641 0.783207 0.621761i \(-0.213582\pi\)
0.783207 + 0.621761i \(0.213582\pi\)
\(510\) −7523.71 −0.653246
\(511\) −1626.03 −0.140766
\(512\) 2217.58 0.191414
\(513\) 1202.25 0.103471
\(514\) 21488.0 1.84396
\(515\) 8048.60 0.688667
\(516\) 1655.90 0.141273
\(517\) −467.929 −0.0398056
\(518\) 1432.09 0.121472
\(519\) 6171.02 0.521922
\(520\) −5846.56 −0.493055
\(521\) −12312.1 −1.03532 −0.517662 0.855585i \(-0.673197\pi\)
−0.517662 + 0.855585i \(0.673197\pi\)
\(522\) 192.924 0.0161763
\(523\) −5106.42 −0.426938 −0.213469 0.976950i \(-0.568476\pi\)
−0.213469 + 0.976950i \(0.568476\pi\)
\(524\) 8609.02 0.717723
\(525\) 369.693 0.0307328
\(526\) −6518.04 −0.540304
\(527\) −29860.4 −2.46819
\(528\) 330.787 0.0272645
\(529\) 6912.83 0.568163
\(530\) −10036.1 −0.822531
\(531\) 531.000 0.0433963
\(532\) −220.279 −0.0179517
\(533\) −7003.98 −0.569186
\(534\) −70.1072 −0.00568134
\(535\) −7783.10 −0.628959
\(536\) 11006.4 0.886947
\(537\) 6586.42 0.529283
\(538\) 15989.1 1.28130
\(539\) −471.362 −0.0376679
\(540\) 580.399 0.0462526
\(541\) 37.9220 0.00301367 0.00150684 0.999999i \(-0.499520\pi\)
0.00150684 + 0.999999i \(0.499520\pi\)
\(542\) −16680.9 −1.32196
\(543\) 3310.43 0.261628
\(544\) 18091.6 1.42586
\(545\) 4696.42 0.369124
\(546\) 902.165 0.0707126
\(547\) −9136.79 −0.714189 −0.357094 0.934068i \(-0.616233\pi\)
−0.357094 + 0.934068i \(0.616233\pi\)
\(548\) −1445.94 −0.112714
\(549\) 5459.72 0.424436
\(550\) −412.127 −0.0319512
\(551\) −281.150 −0.0217376
\(552\) 6294.47 0.485345
\(553\) −204.143 −0.0156981
\(554\) 7092.24 0.543900
\(555\) 5498.91 0.420569
\(556\) −8536.46 −0.651127
\(557\) 3391.09 0.257963 0.128981 0.991647i \(-0.458829\pi\)
0.128981 + 0.991647i \(0.458829\pi\)
\(558\) 7530.13 0.571283
\(559\) 9883.26 0.747795
\(560\) 682.429 0.0514962
\(561\) 502.402 0.0378100
\(562\) −1263.71 −0.0948513
\(563\) −14780.2 −1.10641 −0.553207 0.833044i \(-0.686596\pi\)
−0.553207 + 0.833044i \(0.686596\pi\)
\(564\) 3580.96 0.267350
\(565\) −5439.39 −0.405021
\(566\) −18269.4 −1.35675
\(567\) 113.650 0.00841773
\(568\) 5251.40 0.387930
\(569\) −16176.1 −1.19181 −0.595903 0.803057i \(-0.703206\pi\)
−0.595903 + 0.803057i \(0.703206\pi\)
\(570\) −2764.98 −0.203180
\(571\) −11112.0 −0.814401 −0.407200 0.913339i \(-0.633495\pi\)
−0.407200 + 0.913339i \(0.633495\pi\)
\(572\) −307.654 −0.0224889
\(573\) −2342.55 −0.170788
\(574\) 528.471 0.0384285
\(575\) −12131.7 −0.879875
\(576\) 1181.51 0.0854678
\(577\) 16038.4 1.15717 0.578586 0.815621i \(-0.303605\pi\)
0.578586 + 0.815621i \(0.303605\pi\)
\(578\) 33160.3 2.38631
\(579\) −11471.5 −0.823387
\(580\) −135.728 −0.00971693
\(581\) −1205.03 −0.0860462
\(582\) 5509.28 0.392383
\(583\) 670.171 0.0476083
\(584\) 17603.3 1.24731
\(585\) 3464.11 0.244826
\(586\) 25354.3 1.78733
\(587\) 8129.25 0.571601 0.285801 0.958289i \(-0.407741\pi\)
0.285801 + 0.958289i \(0.407741\pi\)
\(588\) 3607.23 0.252992
\(589\) −10973.8 −0.767684
\(590\) −1221.22 −0.0852147
\(591\) 9437.96 0.656896
\(592\) −23983.8 −1.66508
\(593\) −22305.8 −1.54467 −0.772335 0.635215i \(-0.780912\pi\)
−0.772335 + 0.635215i \(0.780912\pi\)
\(594\) −126.695 −0.00875144
\(595\) 1036.48 0.0714142
\(596\) −3230.66 −0.222035
\(597\) −738.569 −0.0506325
\(598\) −29605.1 −2.02449
\(599\) 1430.54 0.0975797 0.0487898 0.998809i \(-0.484464\pi\)
0.0487898 + 0.998809i \(0.484464\pi\)
\(600\) −4002.28 −0.272321
\(601\) 14630.2 0.992974 0.496487 0.868044i \(-0.334623\pi\)
0.496487 + 0.868044i \(0.334623\pi\)
\(602\) −745.721 −0.0504872
\(603\) −6521.34 −0.440414
\(604\) −8731.84 −0.588234
\(605\) 8103.25 0.544535
\(606\) −14241.9 −0.954680
\(607\) −18345.9 −1.22675 −0.613376 0.789791i \(-0.710189\pi\)
−0.613376 + 0.789791i \(0.710189\pi\)
\(608\) 6648.70 0.443488
\(609\) −26.5775 −0.00176843
\(610\) −12556.5 −0.833439
\(611\) 21373.0 1.41515
\(612\) −3844.77 −0.253947
\(613\) 27920.2 1.83962 0.919808 0.392369i \(-0.128344\pi\)
0.919808 + 0.392369i \(0.128344\pi\)
\(614\) 15193.3 0.998617
\(615\) 2029.21 0.133050
\(616\) −29.4575 −0.00192675
\(617\) 11876.0 0.774892 0.387446 0.921892i \(-0.373357\pi\)
0.387446 + 0.921892i \(0.373357\pi\)
\(618\) 13445.4 0.875164
\(619\) −2811.21 −0.182540 −0.0912699 0.995826i \(-0.529093\pi\)
−0.0912699 + 0.995826i \(0.529093\pi\)
\(620\) −5297.70 −0.343163
\(621\) −3729.50 −0.240998
\(622\) −8430.90 −0.543486
\(623\) 9.65808 0.000621096 0
\(624\) −15108.9 −0.969296
\(625\) 3067.41 0.196314
\(626\) −15873.3 −1.01346
\(627\) 184.634 0.0117601
\(628\) −6671.68 −0.423932
\(629\) −36426.8 −2.30911
\(630\) −261.377 −0.0165294
\(631\) 4558.37 0.287584 0.143792 0.989608i \(-0.454070\pi\)
0.143792 + 0.989608i \(0.454070\pi\)
\(632\) 2210.04 0.139099
\(633\) −1128.97 −0.0708887
\(634\) 17008.6 1.06545
\(635\) −4264.78 −0.266524
\(636\) −5128.67 −0.319756
\(637\) 21529.7 1.33915
\(638\) 29.6281 0.00183854
\(639\) −3111.48 −0.192626
\(640\) −10000.1 −0.617640
\(641\) −2669.67 −0.164502 −0.0822509 0.996612i \(-0.526211\pi\)
−0.0822509 + 0.996612i \(0.526211\pi\)
\(642\) −13001.8 −0.799286
\(643\) 19770.3 1.21254 0.606270 0.795259i \(-0.292665\pi\)
0.606270 + 0.795259i \(0.292665\pi\)
\(644\) 683.329 0.0418120
\(645\) −2863.40 −0.174800
\(646\) 18316.2 1.11555
\(647\) −14551.6 −0.884210 −0.442105 0.896963i \(-0.645768\pi\)
−0.442105 + 0.896963i \(0.645768\pi\)
\(648\) −1230.37 −0.0745887
\(649\) 81.5477 0.00493225
\(650\) 18824.1 1.13591
\(651\) −1037.36 −0.0624538
\(652\) 4443.54 0.266905
\(653\) −19708.2 −1.18108 −0.590538 0.807010i \(-0.701084\pi\)
−0.590538 + 0.807010i \(0.701084\pi\)
\(654\) 7845.46 0.469085
\(655\) −14886.8 −0.888053
\(656\) −8850.51 −0.526760
\(657\) −10430.0 −0.619352
\(658\) −1612.65 −0.0955436
\(659\) 2513.20 0.148559 0.0742794 0.997237i \(-0.476334\pi\)
0.0742794 + 0.997237i \(0.476334\pi\)
\(660\) 89.1341 0.00525688
\(661\) 26382.3 1.55242 0.776212 0.630472i \(-0.217139\pi\)
0.776212 + 0.630472i \(0.217139\pi\)
\(662\) 20613.2 1.21020
\(663\) −22947.5 −1.34421
\(664\) 13045.5 0.762448
\(665\) 380.908 0.0222120
\(666\) 9186.04 0.534462
\(667\) 872.158 0.0506298
\(668\) −9301.94 −0.538777
\(669\) 8968.88 0.518321
\(670\) 14998.0 0.864814
\(671\) 838.471 0.0482396
\(672\) 628.510 0.0360793
\(673\) 21223.6 1.21562 0.607808 0.794084i \(-0.292049\pi\)
0.607808 + 0.794084i \(0.292049\pi\)
\(674\) −1103.41 −0.0630588
\(675\) 2371.37 0.135221
\(676\) 6306.08 0.358789
\(677\) 9297.12 0.527795 0.263898 0.964551i \(-0.414992\pi\)
0.263898 + 0.964551i \(0.414992\pi\)
\(678\) −9086.62 −0.514704
\(679\) −758.966 −0.0428961
\(680\) −11220.9 −0.632795
\(681\) −10788.3 −0.607061
\(682\) 1156.43 0.0649297
\(683\) −19927.0 −1.11638 −0.558188 0.829715i \(-0.688503\pi\)
−0.558188 + 0.829715i \(0.688503\pi\)
\(684\) −1412.96 −0.0789854
\(685\) 2500.33 0.139464
\(686\) −3258.34 −0.181347
\(687\) 3681.21 0.204435
\(688\) 12488.9 0.692055
\(689\) −30610.5 −1.69255
\(690\) 8577.27 0.473233
\(691\) −19009.0 −1.04651 −0.523253 0.852177i \(-0.675282\pi\)
−0.523253 + 0.852177i \(0.675282\pi\)
\(692\) −7252.60 −0.398414
\(693\) 17.4537 0.000956725 0
\(694\) −29378.4 −1.60690
\(695\) 14761.3 0.805652
\(696\) 287.727 0.0156699
\(697\) −13442.2 −0.730503
\(698\) −21070.2 −1.14258
\(699\) 1206.80 0.0653007
\(700\) −434.488 −0.0234602
\(701\) 27125.3 1.46150 0.730749 0.682647i \(-0.239171\pi\)
0.730749 + 0.682647i \(0.239171\pi\)
\(702\) 5786.87 0.311127
\(703\) −13386.9 −0.718205
\(704\) 181.449 0.00971392
\(705\) −6192.22 −0.330798
\(706\) −13739.6 −0.732432
\(707\) 1961.98 0.104368
\(708\) −624.067 −0.0331269
\(709\) −19776.3 −1.04755 −0.523775 0.851856i \(-0.675477\pi\)
−0.523775 + 0.851856i \(0.675477\pi\)
\(710\) 7155.91 0.378249
\(711\) −1309.46 −0.0690697
\(712\) −104.558 −0.00550347
\(713\) 34041.8 1.78804
\(714\) 1731.46 0.0907537
\(715\) 531.997 0.0278260
\(716\) −7740.80 −0.404033
\(717\) 12733.8 0.663252
\(718\) 13006.7 0.676055
\(719\) −11372.9 −0.589900 −0.294950 0.955513i \(-0.595303\pi\)
−0.294950 + 0.955513i \(0.595303\pi\)
\(720\) 4377.39 0.226577
\(721\) −1852.25 −0.0956747
\(722\) −16554.8 −0.853333
\(723\) −14238.0 −0.732387
\(724\) −3890.64 −0.199716
\(725\) −554.554 −0.0284077
\(726\) 13536.6 0.692000
\(727\) −13346.6 −0.680880 −0.340440 0.940266i \(-0.610576\pi\)
−0.340440 + 0.940266i \(0.610576\pi\)
\(728\) 1345.49 0.0684988
\(729\) 729.000 0.0370370
\(730\) 23987.4 1.21619
\(731\) 18968.2 0.959732
\(732\) −6416.63 −0.323997
\(733\) −19528.3 −0.984032 −0.492016 0.870586i \(-0.663740\pi\)
−0.492016 + 0.870586i \(0.663740\pi\)
\(734\) 12804.8 0.643916
\(735\) −6237.64 −0.313033
\(736\) −20625.0 −1.03294
\(737\) −1001.51 −0.0500556
\(738\) 3389.84 0.169081
\(739\) 29281.4 1.45756 0.728779 0.684749i \(-0.240088\pi\)
0.728779 + 0.684749i \(0.240088\pi\)
\(740\) −6462.69 −0.321045
\(741\) −8433.27 −0.418089
\(742\) 2309.65 0.114272
\(743\) 1826.80 0.0902003 0.0451001 0.998982i \(-0.485639\pi\)
0.0451001 + 0.998982i \(0.485639\pi\)
\(744\) 11230.4 0.553398
\(745\) 5586.49 0.274729
\(746\) −18886.6 −0.926927
\(747\) −7729.55 −0.378593
\(748\) −590.457 −0.0288626
\(749\) 1791.15 0.0873795
\(750\) −13215.7 −0.643428
\(751\) 24216.9 1.17668 0.588341 0.808613i \(-0.299781\pi\)
0.588341 + 0.808613i \(0.299781\pi\)
\(752\) 27007.7 1.30967
\(753\) −7533.63 −0.364596
\(754\) −1353.28 −0.0653628
\(755\) 15099.2 0.727834
\(756\) −133.569 −0.00642575
\(757\) −20741.4 −0.995849 −0.497924 0.867221i \(-0.665904\pi\)
−0.497924 + 0.867221i \(0.665904\pi\)
\(758\) 43367.5 2.07807
\(759\) −572.754 −0.0273909
\(760\) −4123.69 −0.196819
\(761\) −15742.5 −0.749890 −0.374945 0.927047i \(-0.622338\pi\)
−0.374945 + 0.927047i \(0.622338\pi\)
\(762\) −7124.40 −0.338700
\(763\) −1080.80 −0.0512813
\(764\) 2753.12 0.130372
\(765\) 6648.41 0.314214
\(766\) −34582.0 −1.63120
\(767\) −3724.74 −0.175349
\(768\) −13554.7 −0.636867
\(769\) −21368.8 −1.00205 −0.501027 0.865432i \(-0.667044\pi\)
−0.501027 + 0.865432i \(0.667044\pi\)
\(770\) −40.1407 −0.00187866
\(771\) −18988.1 −0.886952
\(772\) 13482.1 0.628539
\(773\) −2532.79 −0.117850 −0.0589250 0.998262i \(-0.518767\pi\)
−0.0589250 + 0.998262i \(0.518767\pi\)
\(774\) −4783.37 −0.222138
\(775\) −21645.1 −1.00325
\(776\) 8216.53 0.380098
\(777\) −1265.48 −0.0584285
\(778\) 35174.2 1.62089
\(779\) −4940.05 −0.227209
\(780\) −4071.26 −0.186890
\(781\) −477.842 −0.0218931
\(782\) −56818.9 −2.59826
\(783\) −170.479 −0.00778089
\(784\) 27205.8 1.23933
\(785\) 11536.7 0.524539
\(786\) −24868.7 −1.12854
\(787\) 30214.3 1.36852 0.684259 0.729239i \(-0.260126\pi\)
0.684259 + 0.729239i \(0.260126\pi\)
\(788\) −11092.1 −0.501447
\(789\) 5759.73 0.259888
\(790\) 3011.55 0.135628
\(791\) 1251.79 0.0562685
\(792\) −188.953 −0.00847745
\(793\) −38297.7 −1.71499
\(794\) −4617.18 −0.206370
\(795\) 8868.54 0.395641
\(796\) 868.016 0.0386508
\(797\) 1136.50 0.0505107 0.0252554 0.999681i \(-0.491960\pi\)
0.0252554 + 0.999681i \(0.491960\pi\)
\(798\) 636.315 0.0282272
\(799\) 41019.5 1.81623
\(800\) 13114.2 0.579571
\(801\) 61.9510 0.00273275
\(802\) 7557.91 0.332767
\(803\) −1601.78 −0.0703931
\(804\) 7664.31 0.336193
\(805\) −1181.62 −0.0517348
\(806\) −52820.7 −2.30835
\(807\) −14129.0 −0.616311
\(808\) −21240.3 −0.924791
\(809\) 35914.9 1.56082 0.780408 0.625270i \(-0.215011\pi\)
0.780408 + 0.625270i \(0.215011\pi\)
\(810\) −1676.58 −0.0727274
\(811\) −18539.8 −0.802738 −0.401369 0.915916i \(-0.631466\pi\)
−0.401369 + 0.915916i \(0.631466\pi\)
\(812\) 31.2356 0.00134995
\(813\) 14740.2 0.635870
\(814\) 1410.74 0.0607448
\(815\) −7683.80 −0.330247
\(816\) −28997.4 −1.24401
\(817\) 6970.86 0.298506
\(818\) −12190.3 −0.521056
\(819\) −797.208 −0.0340130
\(820\) −2384.86 −0.101565
\(821\) −2574.44 −0.109438 −0.0547189 0.998502i \(-0.517426\pi\)
−0.0547189 + 0.998502i \(0.517426\pi\)
\(822\) 4176.85 0.177232
\(823\) −35272.7 −1.49396 −0.746980 0.664847i \(-0.768497\pi\)
−0.746980 + 0.664847i \(0.768497\pi\)
\(824\) 20052.4 0.847765
\(825\) 364.180 0.0153687
\(826\) 281.043 0.0118386
\(827\) −24192.0 −1.01722 −0.508609 0.860998i \(-0.669840\pi\)
−0.508609 + 0.860998i \(0.669840\pi\)
\(828\) 4383.16 0.183968
\(829\) 45669.7 1.91336 0.956679 0.291143i \(-0.0940356\pi\)
0.956679 + 0.291143i \(0.0940356\pi\)
\(830\) 17776.7 0.743421
\(831\) −6267.14 −0.261618
\(832\) −8287.77 −0.345345
\(833\) 41320.4 1.71869
\(834\) 24659.1 1.02383
\(835\) 16085.0 0.666640
\(836\) −216.994 −0.00897716
\(837\) −6654.09 −0.274790
\(838\) −29034.2 −1.19686
\(839\) 40744.1 1.67657 0.838285 0.545232i \(-0.183559\pi\)
0.838285 + 0.545232i \(0.183559\pi\)
\(840\) −389.818 −0.0160119
\(841\) −24349.1 −0.998365
\(842\) −45392.8 −1.85788
\(843\) 1116.69 0.0456239
\(844\) 1326.84 0.0541135
\(845\) −10904.5 −0.443937
\(846\) −10344.2 −0.420381
\(847\) −1864.83 −0.0756508
\(848\) −38680.6 −1.56639
\(849\) 16144.0 0.652603
\(850\) 36127.8 1.45785
\(851\) 41527.7 1.67280
\(852\) 3656.82 0.147043
\(853\) 45633.3 1.83172 0.915859 0.401500i \(-0.131511\pi\)
0.915859 + 0.401500i \(0.131511\pi\)
\(854\) 2889.67 0.115787
\(855\) 2443.31 0.0977303
\(856\) −19390.9 −0.774262
\(857\) 34457.3 1.37344 0.686720 0.726922i \(-0.259050\pi\)
0.686720 + 0.726922i \(0.259050\pi\)
\(858\) 888.712 0.0353615
\(859\) 35485.7 1.40950 0.704748 0.709457i \(-0.251060\pi\)
0.704748 + 0.709457i \(0.251060\pi\)
\(860\) 3365.26 0.133435
\(861\) −466.989 −0.0184843
\(862\) 58405.0 2.30775
\(863\) −46492.3 −1.83386 −0.916928 0.399054i \(-0.869339\pi\)
−0.916928 + 0.399054i \(0.869339\pi\)
\(864\) 4031.53 0.158745
\(865\) 12541.2 0.492966
\(866\) 2325.78 0.0912622
\(867\) −29302.5 −1.14782
\(868\) 1219.18 0.0476747
\(869\) −201.099 −0.00785018
\(870\) 392.076 0.0152789
\(871\) 45744.4 1.77955
\(872\) 11700.7 0.454399
\(873\) −4868.33 −0.188738
\(874\) −20881.1 −0.808140
\(875\) 1820.62 0.0703408
\(876\) 12258.1 0.472788
\(877\) −39049.9 −1.50356 −0.751780 0.659414i \(-0.770805\pi\)
−0.751780 + 0.659414i \(0.770805\pi\)
\(878\) −19284.5 −0.741252
\(879\) −22404.6 −0.859714
\(880\) 672.253 0.0257518
\(881\) 43968.2 1.68142 0.840708 0.541489i \(-0.182139\pi\)
0.840708 + 0.541489i \(0.182139\pi\)
\(882\) −10420.1 −0.397804
\(883\) 21974.7 0.837493 0.418746 0.908103i \(-0.362470\pi\)
0.418746 + 0.908103i \(0.362470\pi\)
\(884\) 26969.5 1.02611
\(885\) 1079.14 0.0409886
\(886\) 39575.3 1.50063
\(887\) −8148.07 −0.308439 −0.154220 0.988037i \(-0.549286\pi\)
−0.154220 + 0.988037i \(0.549286\pi\)
\(888\) 13700.1 0.517730
\(889\) 981.468 0.0370274
\(890\) −142.478 −0.00536614
\(891\) 111.955 0.00420948
\(892\) −10540.8 −0.395665
\(893\) 15074.8 0.564903
\(894\) 9332.34 0.349128
\(895\) 13385.5 0.499918
\(896\) 2301.36 0.0858071
\(897\) 26160.9 0.973787
\(898\) 942.139 0.0350107
\(899\) 1556.08 0.0577289
\(900\) −2786.99 −0.103222
\(901\) −58748.4 −2.17225
\(902\) 520.591 0.0192170
\(903\) 658.964 0.0242846
\(904\) −13551.8 −0.498590
\(905\) 6727.72 0.247113
\(906\) 25223.5 0.924937
\(907\) 46795.6 1.71314 0.856572 0.516027i \(-0.172590\pi\)
0.856572 + 0.516027i \(0.172590\pi\)
\(908\) 12679.1 0.463405
\(909\) 12585.0 0.459205
\(910\) 1833.45 0.0667894
\(911\) −11850.0 −0.430964 −0.215482 0.976508i \(-0.569132\pi\)
−0.215482 + 0.976508i \(0.569132\pi\)
\(912\) −10656.6 −0.386925
\(913\) −1187.06 −0.0430294
\(914\) 23312.3 0.843658
\(915\) 11095.7 0.400888
\(916\) −4326.41 −0.156057
\(917\) 3425.94 0.123375
\(918\) 11106.3 0.399306
\(919\) −6238.95 −0.223943 −0.111972 0.993711i \(-0.535717\pi\)
−0.111972 + 0.993711i \(0.535717\pi\)
\(920\) 12792.1 0.458418
\(921\) −13425.7 −0.480339
\(922\) −30720.9 −1.09733
\(923\) 21825.7 0.778335
\(924\) −20.5127 −0.000730324 0
\(925\) −26405.0 −0.938584
\(926\) 51852.4 1.84015
\(927\) −11881.1 −0.420958
\(928\) −942.788 −0.0333497
\(929\) 8394.89 0.296477 0.148239 0.988952i \(-0.452640\pi\)
0.148239 + 0.988952i \(0.452640\pi\)
\(930\) 15303.3 0.539588
\(931\) 15185.4 0.534565
\(932\) −1418.31 −0.0498478
\(933\) 7450.06 0.261419
\(934\) −53382.3 −1.87015
\(935\) 1021.02 0.0357123
\(936\) 8630.53 0.301387
\(937\) −45451.4 −1.58467 −0.792333 0.610089i \(-0.791134\pi\)
−0.792333 + 0.610089i \(0.791134\pi\)
\(938\) −3451.55 −0.120146
\(939\) 14026.6 0.487476
\(940\) 7277.52 0.252517
\(941\) −6882.68 −0.238437 −0.119218 0.992868i \(-0.538039\pi\)
−0.119218 + 0.992868i \(0.538039\pi\)
\(942\) 19272.3 0.666589
\(943\) 15324.6 0.529201
\(944\) −4706.73 −0.162279
\(945\) 230.969 0.00795071
\(946\) −734.601 −0.0252473
\(947\) −54450.8 −1.86844 −0.934220 0.356698i \(-0.883903\pi\)
−0.934220 + 0.356698i \(0.883903\pi\)
\(948\) 1538.96 0.0527249
\(949\) 73162.4 2.50258
\(950\) 13277.1 0.453436
\(951\) −15029.8 −0.512488
\(952\) 2582.29 0.0879124
\(953\) −1727.06 −0.0587041 −0.0293521 0.999569i \(-0.509344\pi\)
−0.0293521 + 0.999569i \(0.509344\pi\)
\(954\) 14815.1 0.502784
\(955\) −4760.72 −0.161312
\(956\) −14965.6 −0.506299
\(957\) −26.1812 −0.000884344 0
\(958\) −204.690 −0.00690317
\(959\) −575.409 −0.0193753
\(960\) 2401.15 0.0807259
\(961\) 30945.4 1.03875
\(962\) −64436.3 −2.15957
\(963\) 11489.2 0.384460
\(964\) 16733.4 0.559074
\(965\) −23313.4 −0.777705
\(966\) −1973.92 −0.0657450
\(967\) −22506.8 −0.748468 −0.374234 0.927334i \(-0.622094\pi\)
−0.374234 + 0.927334i \(0.622094\pi\)
\(968\) 20188.5 0.670335
\(969\) −16185.4 −0.536583
\(970\) 11196.4 0.370613
\(971\) 55203.2 1.82446 0.912232 0.409673i \(-0.134357\pi\)
0.912232 + 0.409673i \(0.134357\pi\)
\(972\) −856.770 −0.0282725
\(973\) −3397.07 −0.111927
\(974\) 25683.1 0.844906
\(975\) −16634.2 −0.546379
\(976\) −48394.4 −1.58716
\(977\) −26202.6 −0.858029 −0.429015 0.903298i \(-0.641139\pi\)
−0.429015 + 0.903298i \(0.641139\pi\)
\(978\) −12835.9 −0.419681
\(979\) 9.51406 0.000310593 0
\(980\) 7330.90 0.238956
\(981\) −6932.73 −0.225632
\(982\) 5672.71 0.184342
\(983\) −28303.1 −0.918342 −0.459171 0.888348i \(-0.651853\pi\)
−0.459171 + 0.888348i \(0.651853\pi\)
\(984\) 5055.60 0.163787
\(985\) 19180.6 0.620451
\(986\) −2597.25 −0.0838877
\(987\) 1425.04 0.0459569
\(988\) 9911.35 0.319152
\(989\) −21624.4 −0.695262
\(990\) −257.480 −0.00826590
\(991\) 51816.8 1.66096 0.830482 0.557046i \(-0.188065\pi\)
0.830482 + 0.557046i \(0.188065\pi\)
\(992\) −36798.6 −1.17778
\(993\) −18215.1 −0.582112
\(994\) −1646.81 −0.0525491
\(995\) −1500.98 −0.0478234
\(996\) 9084.28 0.289002
\(997\) 37249.5 1.18325 0.591627 0.806212i \(-0.298486\pi\)
0.591627 + 0.806212i \(0.298486\pi\)
\(998\) −13989.4 −0.443715
\(999\) −8117.35 −0.257079
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 177.4.a.b.1.6 7
3.2 odd 2 531.4.a.c.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.b.1.6 7 1.1 even 1 trivial
531.4.a.c.1.2 7 3.2 odd 2