Properties

Label 177.6.a.b.1.7
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.501674\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.501674 q^{2} -9.00000 q^{3} -31.7483 q^{4} -14.0506 q^{5} +4.51506 q^{6} +117.234 q^{7} +31.9809 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-0.501674 q^{2} -9.00000 q^{3} -31.7483 q^{4} -14.0506 q^{5} +4.51506 q^{6} +117.234 q^{7} +31.9809 q^{8} +81.0000 q^{9} +7.04881 q^{10} +53.5537 q^{11} +285.735 q^{12} -471.768 q^{13} -58.8130 q^{14} +126.455 q^{15} +999.902 q^{16} +1164.87 q^{17} -40.6356 q^{18} +1826.16 q^{19} +446.082 q^{20} -1055.10 q^{21} -26.8665 q^{22} +254.207 q^{23} -287.828 q^{24} -2927.58 q^{25} +236.673 q^{26} -729.000 q^{27} -3721.97 q^{28} -6598.12 q^{29} -63.4393 q^{30} +1104.44 q^{31} -1525.01 q^{32} -481.983 q^{33} -584.382 q^{34} -1647.20 q^{35} -2571.61 q^{36} -7748.51 q^{37} -916.136 q^{38} +4245.91 q^{39} -449.350 q^{40} -5824.87 q^{41} +529.317 q^{42} -6404.30 q^{43} -1700.24 q^{44} -1138.10 q^{45} -127.529 q^{46} -2856.66 q^{47} -8999.12 q^{48} -3063.28 q^{49} +1468.69 q^{50} -10483.8 q^{51} +14977.8 q^{52} -16675.9 q^{53} +365.720 q^{54} -752.461 q^{55} +3749.23 q^{56} -16435.4 q^{57} +3310.10 q^{58} -3481.00 q^{59} -4014.74 q^{60} +37099.5 q^{61} -554.070 q^{62} +9495.92 q^{63} -31231.8 q^{64} +6628.61 q^{65} +241.798 q^{66} +10974.2 q^{67} -36982.5 q^{68} -2287.86 q^{69} +826.357 q^{70} -15478.9 q^{71} +2590.45 q^{72} -42171.4 q^{73} +3887.22 q^{74} +26348.2 q^{75} -57977.5 q^{76} +6278.29 q^{77} -2130.06 q^{78} -89645.3 q^{79} -14049.2 q^{80} +6561.00 q^{81} +2922.18 q^{82} +58689.8 q^{83} +33497.7 q^{84} -16367.0 q^{85} +3212.87 q^{86} +59383.1 q^{87} +1712.69 q^{88} +61752.2 q^{89} +570.953 q^{90} -55307.0 q^{91} -8070.64 q^{92} -9939.99 q^{93} +1433.11 q^{94} -25658.6 q^{95} +13725.1 q^{96} -13874.8 q^{97} +1536.77 q^{98} +4337.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.501674 −0.0886842 −0.0443421 0.999016i \(-0.514119\pi\)
−0.0443421 + 0.999016i \(0.514119\pi\)
\(3\) −9.00000 −0.577350
\(4\) −31.7483 −0.992135
\(5\) −14.0506 −0.251344 −0.125672 0.992072i \(-0.540109\pi\)
−0.125672 + 0.992072i \(0.540109\pi\)
\(6\) 4.51506 0.0512019
\(7\) 117.234 0.904288 0.452144 0.891945i \(-0.350659\pi\)
0.452144 + 0.891945i \(0.350659\pi\)
\(8\) 31.9809 0.176671
\(9\) 81.0000 0.333333
\(10\) 7.04881 0.0222903
\(11\) 53.5537 0.133447 0.0667233 0.997772i \(-0.478746\pi\)
0.0667233 + 0.997772i \(0.478746\pi\)
\(12\) 285.735 0.572809
\(13\) −471.768 −0.774230 −0.387115 0.922031i \(-0.626528\pi\)
−0.387115 + 0.922031i \(0.626528\pi\)
\(14\) −58.8130 −0.0801961
\(15\) 126.455 0.145114
\(16\) 999.902 0.976467
\(17\) 1164.87 0.977582 0.488791 0.872401i \(-0.337438\pi\)
0.488791 + 0.872401i \(0.337438\pi\)
\(18\) −40.6356 −0.0295614
\(19\) 1826.16 1.16053 0.580263 0.814429i \(-0.302950\pi\)
0.580263 + 0.814429i \(0.302950\pi\)
\(20\) 446.082 0.249368
\(21\) −1055.10 −0.522091
\(22\) −26.8665 −0.0118346
\(23\) 254.207 0.100200 0.0501000 0.998744i \(-0.484046\pi\)
0.0501000 + 0.998744i \(0.484046\pi\)
\(24\) −287.828 −0.102001
\(25\) −2927.58 −0.936826
\(26\) 236.673 0.0686620
\(27\) −729.000 −0.192450
\(28\) −3721.97 −0.897176
\(29\) −6598.12 −1.45689 −0.728443 0.685107i \(-0.759755\pi\)
−0.728443 + 0.685107i \(0.759755\pi\)
\(30\) −63.4393 −0.0128693
\(31\) 1104.44 0.206414 0.103207 0.994660i \(-0.467090\pi\)
0.103207 + 0.994660i \(0.467090\pi\)
\(32\) −1525.01 −0.263268
\(33\) −481.983 −0.0770455
\(34\) −584.382 −0.0866961
\(35\) −1647.20 −0.227288
\(36\) −2571.61 −0.330712
\(37\) −7748.51 −0.930494 −0.465247 0.885181i \(-0.654035\pi\)
−0.465247 + 0.885181i \(0.654035\pi\)
\(38\) −916.136 −0.102920
\(39\) 4245.91 0.447002
\(40\) −449.350 −0.0444053
\(41\) −5824.87 −0.541161 −0.270581 0.962697i \(-0.587216\pi\)
−0.270581 + 0.962697i \(0.587216\pi\)
\(42\) 529.317 0.0463012
\(43\) −6404.30 −0.528203 −0.264101 0.964495i \(-0.585075\pi\)
−0.264101 + 0.964495i \(0.585075\pi\)
\(44\) −1700.24 −0.132397
\(45\) −1138.10 −0.0837815
\(46\) −127.529 −0.00888616
\(47\) −2856.66 −0.188632 −0.0943158 0.995542i \(-0.530066\pi\)
−0.0943158 + 0.995542i \(0.530066\pi\)
\(48\) −8999.12 −0.563764
\(49\) −3063.28 −0.182262
\(50\) 1468.69 0.0830817
\(51\) −10483.8 −0.564407
\(52\) 14977.8 0.768141
\(53\) −16675.9 −0.815452 −0.407726 0.913104i \(-0.633678\pi\)
−0.407726 + 0.913104i \(0.633678\pi\)
\(54\) 365.720 0.0170673
\(55\) −752.461 −0.0335411
\(56\) 3749.23 0.159761
\(57\) −16435.4 −0.670030
\(58\) 3310.10 0.129203
\(59\) −3481.00 −0.130189
\(60\) −4014.74 −0.143972
\(61\) 37099.5 1.27657 0.638284 0.769801i \(-0.279645\pi\)
0.638284 + 0.769801i \(0.279645\pi\)
\(62\) −554.070 −0.0183057
\(63\) 9495.92 0.301429
\(64\) −31231.8 −0.953119
\(65\) 6628.61 0.194598
\(66\) 241.798 0.00683272
\(67\) 10974.2 0.298665 0.149332 0.988787i \(-0.452288\pi\)
0.149332 + 0.988787i \(0.452288\pi\)
\(68\) −36982.5 −0.969894
\(69\) −2287.86 −0.0578505
\(70\) 826.357 0.0201568
\(71\) −15478.9 −0.364414 −0.182207 0.983260i \(-0.558324\pi\)
−0.182207 + 0.983260i \(0.558324\pi\)
\(72\) 2590.45 0.0588903
\(73\) −42171.4 −0.926214 −0.463107 0.886302i \(-0.653265\pi\)
−0.463107 + 0.886302i \(0.653265\pi\)
\(74\) 3887.22 0.0825202
\(75\) 26348.2 0.540877
\(76\) −57977.5 −1.15140
\(77\) 6278.29 0.120674
\(78\) −2130.06 −0.0396420
\(79\) −89645.3 −1.61607 −0.808034 0.589136i \(-0.799468\pi\)
−0.808034 + 0.589136i \(0.799468\pi\)
\(80\) −14049.2 −0.245430
\(81\) 6561.00 0.111111
\(82\) 2922.18 0.0479924
\(83\) 58689.8 0.935120 0.467560 0.883961i \(-0.345133\pi\)
0.467560 + 0.883961i \(0.345133\pi\)
\(84\) 33497.7 0.517985
\(85\) −16367.0 −0.245710
\(86\) 3212.87 0.0468433
\(87\) 59383.1 0.841133
\(88\) 1712.69 0.0235761
\(89\) 61752.2 0.826376 0.413188 0.910646i \(-0.364415\pi\)
0.413188 + 0.910646i \(0.364415\pi\)
\(90\) 570.953 0.00743010
\(91\) −55307.0 −0.700127
\(92\) −8070.64 −0.0994119
\(93\) −9939.99 −0.119173
\(94\) 1433.11 0.0167286
\(95\) −25658.6 −0.291692
\(96\) 13725.1 0.151998
\(97\) −13874.8 −0.149726 −0.0748630 0.997194i \(-0.523852\pi\)
−0.0748630 + 0.997194i \(0.523852\pi\)
\(98\) 1536.77 0.0161638
\(99\) 4337.85 0.0444822
\(100\) 92945.8 0.929458
\(101\) 101934. 0.994293 0.497146 0.867667i \(-0.334381\pi\)
0.497146 + 0.867667i \(0.334381\pi\)
\(102\) 5259.44 0.0500540
\(103\) −176131. −1.63585 −0.817925 0.575325i \(-0.804875\pi\)
−0.817925 + 0.575325i \(0.804875\pi\)
\(104\) −15087.5 −0.136784
\(105\) 14824.8 0.131225
\(106\) 8365.84 0.0723177
\(107\) 140675. 1.18783 0.593917 0.804526i \(-0.297581\pi\)
0.593917 + 0.804526i \(0.297581\pi\)
\(108\) 23144.5 0.190936
\(109\) −87074.8 −0.701982 −0.350991 0.936379i \(-0.614155\pi\)
−0.350991 + 0.936379i \(0.614155\pi\)
\(110\) 377.490 0.00297456
\(111\) 69736.6 0.537221
\(112\) 117222. 0.883008
\(113\) −257288. −1.89550 −0.947748 0.319019i \(-0.896647\pi\)
−0.947748 + 0.319019i \(0.896647\pi\)
\(114\) 8245.23 0.0594211
\(115\) −3571.75 −0.0251847
\(116\) 209479. 1.44543
\(117\) −38213.2 −0.258077
\(118\) 1746.33 0.0115457
\(119\) 136561. 0.884016
\(120\) 4044.15 0.0256374
\(121\) −158183. −0.982192
\(122\) −18611.9 −0.113211
\(123\) 52423.8 0.312439
\(124\) −35064.2 −0.204791
\(125\) 85042.3 0.486810
\(126\) −4763.85 −0.0267320
\(127\) −60712.0 −0.334014 −0.167007 0.985956i \(-0.553410\pi\)
−0.167007 + 0.985956i \(0.553410\pi\)
\(128\) 64468.6 0.347795
\(129\) 57638.7 0.304958
\(130\) −3325.40 −0.0172578
\(131\) 191296. 0.973928 0.486964 0.873422i \(-0.338104\pi\)
0.486964 + 0.873422i \(0.338104\pi\)
\(132\) 15302.2 0.0764395
\(133\) 214087. 1.04945
\(134\) −5505.44 −0.0264869
\(135\) 10242.9 0.0483713
\(136\) 37253.4 0.172710
\(137\) 262613. 1.19541 0.597703 0.801718i \(-0.296080\pi\)
0.597703 + 0.801718i \(0.296080\pi\)
\(138\) 1147.76 0.00513043
\(139\) −149545. −0.656502 −0.328251 0.944590i \(-0.606459\pi\)
−0.328251 + 0.944590i \(0.606459\pi\)
\(140\) 52295.8 0.225500
\(141\) 25710.0 0.108906
\(142\) 7765.38 0.0323178
\(143\) −25264.9 −0.103318
\(144\) 80992.1 0.325489
\(145\) 92707.4 0.366180
\(146\) 21156.3 0.0821406
\(147\) 27569.6 0.105229
\(148\) 246002. 0.923176
\(149\) −356966. −1.31723 −0.658614 0.752481i \(-0.728857\pi\)
−0.658614 + 0.752481i \(0.728857\pi\)
\(150\) −13218.2 −0.0479672
\(151\) −236793. −0.845135 −0.422568 0.906331i \(-0.638871\pi\)
−0.422568 + 0.906331i \(0.638871\pi\)
\(152\) 58402.2 0.205031
\(153\) 94354.1 0.325861
\(154\) −3149.65 −0.0107019
\(155\) −15518.1 −0.0518810
\(156\) −134801. −0.443486
\(157\) 13948.1 0.0451613 0.0225807 0.999745i \(-0.492812\pi\)
0.0225807 + 0.999745i \(0.492812\pi\)
\(158\) 44972.7 0.143320
\(159\) 150083. 0.470802
\(160\) 21427.3 0.0661710
\(161\) 29801.6 0.0906097
\(162\) −3291.48 −0.00985380
\(163\) 119894. 0.353450 0.176725 0.984260i \(-0.443450\pi\)
0.176725 + 0.984260i \(0.443450\pi\)
\(164\) 184930. 0.536905
\(165\) 6772.15 0.0193650
\(166\) −29443.1 −0.0829304
\(167\) −6906.10 −0.0191620 −0.00958102 0.999954i \(-0.503050\pi\)
−0.00958102 + 0.999954i \(0.503050\pi\)
\(168\) −33743.1 −0.0922383
\(169\) −148728. −0.400568
\(170\) 8210.91 0.0217906
\(171\) 147919. 0.386842
\(172\) 203326. 0.524049
\(173\) −394688. −1.00263 −0.501313 0.865266i \(-0.667149\pi\)
−0.501313 + 0.865266i \(0.667149\pi\)
\(174\) −29790.9 −0.0745952
\(175\) −343211. −0.847161
\(176\) 53548.5 0.130306
\(177\) 31329.0 0.0751646
\(178\) −30979.5 −0.0732865
\(179\) 728867. 1.70026 0.850130 0.526572i \(-0.176523\pi\)
0.850130 + 0.526572i \(0.176523\pi\)
\(180\) 36132.7 0.0831225
\(181\) −409033. −0.928030 −0.464015 0.885827i \(-0.653592\pi\)
−0.464015 + 0.885827i \(0.653592\pi\)
\(182\) 27746.1 0.0620902
\(183\) −333896. −0.737026
\(184\) 8129.75 0.0177024
\(185\) 108871. 0.233875
\(186\) 4986.63 0.0105688
\(187\) 62382.8 0.130455
\(188\) 90694.3 0.187148
\(189\) −85463.3 −0.174030
\(190\) 12872.2 0.0258685
\(191\) −549926. −1.09074 −0.545369 0.838196i \(-0.683611\pi\)
−0.545369 + 0.838196i \(0.683611\pi\)
\(192\) 281086. 0.550284
\(193\) 272527. 0.526644 0.263322 0.964708i \(-0.415182\pi\)
0.263322 + 0.964708i \(0.415182\pi\)
\(194\) 6960.61 0.0132783
\(195\) −59657.5 −0.112351
\(196\) 97254.1 0.180829
\(197\) −196950. −0.361569 −0.180784 0.983523i \(-0.557864\pi\)
−0.180784 + 0.983523i \(0.557864\pi\)
\(198\) −2176.18 −0.00394487
\(199\) −573962. −1.02743 −0.513713 0.857962i \(-0.671730\pi\)
−0.513713 + 0.857962i \(0.671730\pi\)
\(200\) −93626.6 −0.165510
\(201\) −98767.4 −0.172434
\(202\) −51137.4 −0.0881781
\(203\) −773522. −1.31744
\(204\) 332843. 0.559968
\(205\) 81842.8 0.136018
\(206\) 88360.4 0.145074
\(207\) 20590.8 0.0334000
\(208\) −471722. −0.756010
\(209\) 97797.6 0.154868
\(210\) −7437.21 −0.0116376
\(211\) −812952. −1.25707 −0.628534 0.777782i \(-0.716345\pi\)
−0.628534 + 0.777782i \(0.716345\pi\)
\(212\) 529431. 0.809039
\(213\) 139311. 0.210395
\(214\) −70572.7 −0.105342
\(215\) 89984.2 0.132761
\(216\) −23314.0 −0.0340003
\(217\) 129478. 0.186658
\(218\) 43683.1 0.0622548
\(219\) 379543. 0.534750
\(220\) 23889.4 0.0332773
\(221\) −549546. −0.756873
\(222\) −34985.0 −0.0476430
\(223\) −211386. −0.284652 −0.142326 0.989820i \(-0.545458\pi\)
−0.142326 + 0.989820i \(0.545458\pi\)
\(224\) −178783. −0.238070
\(225\) −237134. −0.312275
\(226\) 129074. 0.168101
\(227\) −383490. −0.493958 −0.246979 0.969021i \(-0.579438\pi\)
−0.246979 + 0.969021i \(0.579438\pi\)
\(228\) 521798. 0.664760
\(229\) −67560.6 −0.0851344 −0.0425672 0.999094i \(-0.513554\pi\)
−0.0425672 + 0.999094i \(0.513554\pi\)
\(230\) 1791.86 0.00223349
\(231\) −56504.6 −0.0696713
\(232\) −211014. −0.257389
\(233\) 592445. 0.714922 0.357461 0.933928i \(-0.383643\pi\)
0.357461 + 0.933928i \(0.383643\pi\)
\(234\) 19170.6 0.0228873
\(235\) 40137.8 0.0474115
\(236\) 110516. 0.129165
\(237\) 806807. 0.933037
\(238\) −68509.2 −0.0783983
\(239\) 45816.0 0.0518827 0.0259413 0.999663i \(-0.491742\pi\)
0.0259413 + 0.999663i \(0.491742\pi\)
\(240\) 126443. 0.141699
\(241\) −1.56926e6 −1.74041 −0.870204 0.492691i \(-0.836013\pi\)
−0.870204 + 0.492691i \(0.836013\pi\)
\(242\) 79356.3 0.0871049
\(243\) −59049.0 −0.0641500
\(244\) −1.17785e6 −1.26653
\(245\) 43040.9 0.0458106
\(246\) −26299.7 −0.0277085
\(247\) −861523. −0.898514
\(248\) 35321.0 0.0364673
\(249\) −528208. −0.539892
\(250\) −42663.5 −0.0431724
\(251\) −36553.8 −0.0366225 −0.0183113 0.999832i \(-0.505829\pi\)
−0.0183113 + 0.999832i \(0.505829\pi\)
\(252\) −301480. −0.299059
\(253\) 13613.7 0.0133714
\(254\) 30457.6 0.0296218
\(255\) 147303. 0.141861
\(256\) 967076. 0.922276
\(257\) −821929. −0.776250 −0.388125 0.921607i \(-0.626877\pi\)
−0.388125 + 0.921607i \(0.626877\pi\)
\(258\) −28915.8 −0.0270450
\(259\) −908385. −0.841435
\(260\) −210447. −0.193068
\(261\) −534448. −0.485628
\(262\) −95968.0 −0.0863721
\(263\) −242612. −0.216283 −0.108141 0.994136i \(-0.534490\pi\)
−0.108141 + 0.994136i \(0.534490\pi\)
\(264\) −15414.2 −0.0136117
\(265\) 234306. 0.204959
\(266\) −107402. −0.0930697
\(267\) −555770. −0.477108
\(268\) −348411. −0.296316
\(269\) −1.06646e6 −0.898596 −0.449298 0.893382i \(-0.648326\pi\)
−0.449298 + 0.893382i \(0.648326\pi\)
\(270\) −5138.58 −0.00428977
\(271\) 202507. 0.167501 0.0837505 0.996487i \(-0.473310\pi\)
0.0837505 + 0.996487i \(0.473310\pi\)
\(272\) 1.16475e6 0.954577
\(273\) 497763. 0.404219
\(274\) −131746. −0.106014
\(275\) −156783. −0.125016
\(276\) 72635.8 0.0573955
\(277\) 1.26102e6 0.987469 0.493735 0.869613i \(-0.335631\pi\)
0.493735 + 0.869613i \(0.335631\pi\)
\(278\) 75023.0 0.0582214
\(279\) 89459.9 0.0688047
\(280\) −52678.9 −0.0401552
\(281\) 1.93346e6 1.46072 0.730362 0.683060i \(-0.239351\pi\)
0.730362 + 0.683060i \(0.239351\pi\)
\(282\) −12898.0 −0.00965829
\(283\) −502495. −0.372963 −0.186482 0.982458i \(-0.559708\pi\)
−0.186482 + 0.982458i \(0.559708\pi\)
\(284\) 491431. 0.361548
\(285\) 230927. 0.168408
\(286\) 12674.7 0.00916271
\(287\) −682870. −0.489366
\(288\) −123526. −0.0877561
\(289\) −62946.4 −0.0443329
\(290\) −46508.9 −0.0324744
\(291\) 124873. 0.0864443
\(292\) 1.33887e6 0.918929
\(293\) −86625.1 −0.0589487 −0.0294744 0.999566i \(-0.509383\pi\)
−0.0294744 + 0.999566i \(0.509383\pi\)
\(294\) −13830.9 −0.00933217
\(295\) 48910.1 0.0327223
\(296\) −247804. −0.164391
\(297\) −39040.6 −0.0256818
\(298\) 179080. 0.116817
\(299\) −119927. −0.0775778
\(300\) −836512. −0.536623
\(301\) −750800. −0.477648
\(302\) 118793. 0.0749502
\(303\) −917403. −0.574055
\(304\) 1.82598e6 1.13322
\(305\) −521270. −0.320858
\(306\) −47335.0 −0.0288987
\(307\) 630671. 0.381907 0.190953 0.981599i \(-0.438842\pi\)
0.190953 + 0.981599i \(0.438842\pi\)
\(308\) −199325. −0.119725
\(309\) 1.58518e6 0.944458
\(310\) 7785.01 0.00460103
\(311\) −828845. −0.485929 −0.242964 0.970035i \(-0.578120\pi\)
−0.242964 + 0.970035i \(0.578120\pi\)
\(312\) 135788. 0.0789722
\(313\) 1.64437e6 0.948724 0.474362 0.880330i \(-0.342679\pi\)
0.474362 + 0.880330i \(0.342679\pi\)
\(314\) −6997.40 −0.00400510
\(315\) −133423. −0.0757626
\(316\) 2.84609e6 1.60336
\(317\) 2.66630e6 1.49025 0.745127 0.666923i \(-0.232389\pi\)
0.745127 + 0.666923i \(0.232389\pi\)
\(318\) −75292.6 −0.0417527
\(319\) −353354. −0.194416
\(320\) 438825. 0.239561
\(321\) −1.26607e6 −0.685797
\(322\) −14950.7 −0.00803565
\(323\) 2.12723e6 1.13451
\(324\) −208301. −0.110237
\(325\) 1.38114e6 0.725319
\(326\) −60147.6 −0.0313454
\(327\) 783673. 0.405290
\(328\) −186284. −0.0956074
\(329\) −334897. −0.170577
\(330\) −3397.41 −0.00171737
\(331\) 1.09024e6 0.546955 0.273477 0.961878i \(-0.411826\pi\)
0.273477 + 0.961878i \(0.411826\pi\)
\(332\) −1.86330e6 −0.927765
\(333\) −627629. −0.310165
\(334\) 3464.61 0.00169937
\(335\) −154193. −0.0750677
\(336\) −1.05500e6 −0.509805
\(337\) −1.37111e6 −0.657654 −0.328827 0.944390i \(-0.606653\pi\)
−0.328827 + 0.944390i \(0.606653\pi\)
\(338\) 74613.0 0.0355241
\(339\) 2.31559e6 1.09437
\(340\) 519626. 0.243777
\(341\) 59147.0 0.0275453
\(342\) −74207.0 −0.0343068
\(343\) −2.32946e6 −1.06911
\(344\) −204815. −0.0933181
\(345\) 32145.8 0.0145404
\(346\) 198005. 0.0889170
\(347\) −2.35316e6 −1.04913 −0.524564 0.851371i \(-0.675772\pi\)
−0.524564 + 0.851371i \(0.675772\pi\)
\(348\) −1.88531e6 −0.834518
\(349\) 844331. 0.371064 0.185532 0.982638i \(-0.440599\pi\)
0.185532 + 0.982638i \(0.440599\pi\)
\(350\) 172180. 0.0751298
\(351\) 343919. 0.149001
\(352\) −81670.0 −0.0351323
\(353\) 1.07476e6 0.459066 0.229533 0.973301i \(-0.426280\pi\)
0.229533 + 0.973301i \(0.426280\pi\)
\(354\) −15716.9 −0.00666591
\(355\) 217488. 0.0915935
\(356\) −1.96053e6 −0.819877
\(357\) −1.22905e6 −0.510387
\(358\) −365653. −0.150786
\(359\) 1.95175e6 0.799259 0.399629 0.916677i \(-0.369139\pi\)
0.399629 + 0.916677i \(0.369139\pi\)
\(360\) −36397.3 −0.0148018
\(361\) 858761. 0.346820
\(362\) 205201. 0.0823016
\(363\) 1.42365e6 0.567069
\(364\) 1.75591e6 0.694621
\(365\) 592533. 0.232799
\(366\) 167507. 0.0653626
\(367\) 1.56632e6 0.607036 0.303518 0.952826i \(-0.401839\pi\)
0.303518 + 0.952826i \(0.401839\pi\)
\(368\) 254182. 0.0978420
\(369\) −471814. −0.180387
\(370\) −54617.7 −0.0207410
\(371\) −1.95497e6 −0.737404
\(372\) 315578. 0.118236
\(373\) 2.85853e6 1.06382 0.531912 0.846799i \(-0.321474\pi\)
0.531912 + 0.846799i \(0.321474\pi\)
\(374\) −31295.8 −0.0115693
\(375\) −765381. −0.281060
\(376\) −91358.5 −0.0333257
\(377\) 3.11278e6 1.12796
\(378\) 42874.7 0.0154337
\(379\) 1.66138e6 0.594116 0.297058 0.954859i \(-0.403995\pi\)
0.297058 + 0.954859i \(0.403995\pi\)
\(380\) 814618. 0.289398
\(381\) 546408. 0.192843
\(382\) 275883. 0.0967313
\(383\) −4.91425e6 −1.71183 −0.855914 0.517118i \(-0.827005\pi\)
−0.855914 + 0.517118i \(0.827005\pi\)
\(384\) −580217. −0.200799
\(385\) −88213.7 −0.0303308
\(386\) −136720. −0.0467050
\(387\) −518749. −0.176068
\(388\) 440501. 0.148548
\(389\) 4.04295e6 1.35464 0.677320 0.735688i \(-0.263141\pi\)
0.677320 + 0.735688i \(0.263141\pi\)
\(390\) 29928.6 0.00996380
\(391\) 296117. 0.0979538
\(392\) −97966.4 −0.0322005
\(393\) −1.72166e6 −0.562298
\(394\) 98804.7 0.0320654
\(395\) 1.25957e6 0.406190
\(396\) −137719. −0.0441324
\(397\) 2.31593e6 0.737477 0.368738 0.929533i \(-0.379790\pi\)
0.368738 + 0.929533i \(0.379790\pi\)
\(398\) 287942. 0.0911164
\(399\) −1.92679e6 −0.605900
\(400\) −2.92730e6 −0.914780
\(401\) −3.24006e6 −1.00622 −0.503110 0.864223i \(-0.667811\pi\)
−0.503110 + 0.864223i \(0.667811\pi\)
\(402\) 49549.0 0.0152922
\(403\) −521041. −0.159812
\(404\) −3.23622e6 −0.986473
\(405\) −92185.9 −0.0279272
\(406\) 388055. 0.116837
\(407\) −414961. −0.124171
\(408\) −335280. −0.0997144
\(409\) 2.30082e6 0.680101 0.340051 0.940407i \(-0.389556\pi\)
0.340051 + 0.940407i \(0.389556\pi\)
\(410\) −41058.4 −0.0120626
\(411\) −2.36352e6 −0.690168
\(412\) 5.59187e6 1.62298
\(413\) −408090. −0.117728
\(414\) −10329.8 −0.00296205
\(415\) −824626. −0.235037
\(416\) 719452. 0.203830
\(417\) 1.34591e6 0.379032
\(418\) −49062.5 −0.0137344
\(419\) 4.71896e6 1.31314 0.656570 0.754265i \(-0.272007\pi\)
0.656570 + 0.754265i \(0.272007\pi\)
\(420\) −470663. −0.130193
\(421\) 1.59180e6 0.437708 0.218854 0.975758i \(-0.429768\pi\)
0.218854 + 0.975758i \(0.429768\pi\)
\(422\) 407837. 0.111482
\(423\) −231390. −0.0628772
\(424\) −533308. −0.144067
\(425\) −3.41024e6 −0.915824
\(426\) −69888.4 −0.0186587
\(427\) 4.34931e6 1.15438
\(428\) −4.46618e6 −1.17849
\(429\) 227384. 0.0596509
\(430\) −45142.7 −0.0117738
\(431\) −3.84179e6 −0.996186 −0.498093 0.867124i \(-0.665966\pi\)
−0.498093 + 0.867124i \(0.665966\pi\)
\(432\) −728929. −0.187921
\(433\) −97620.0 −0.0250218 −0.0125109 0.999922i \(-0.503982\pi\)
−0.0125109 + 0.999922i \(0.503982\pi\)
\(434\) −64955.6 −0.0165536
\(435\) −834367. −0.211414
\(436\) 2.76448e6 0.696461
\(437\) 464222. 0.116285
\(438\) −190407. −0.0474239
\(439\) 4.43043e6 1.09720 0.548599 0.836086i \(-0.315161\pi\)
0.548599 + 0.836086i \(0.315161\pi\)
\(440\) −24064.3 −0.00592573
\(441\) −248126. −0.0607541
\(442\) 275693. 0.0671227
\(443\) −6.14299e6 −1.48720 −0.743602 0.668623i \(-0.766884\pi\)
−0.743602 + 0.668623i \(0.766884\pi\)
\(444\) −2.21402e6 −0.532996
\(445\) −867655. −0.207705
\(446\) 106047. 0.0252441
\(447\) 3.21269e6 0.760502
\(448\) −3.66142e6 −0.861895
\(449\) 5.39476e6 1.26286 0.631431 0.775432i \(-0.282468\pi\)
0.631431 + 0.775432i \(0.282468\pi\)
\(450\) 118964. 0.0276939
\(451\) −311943. −0.0722161
\(452\) 8.16845e6 1.88059
\(453\) 2.13114e6 0.487939
\(454\) 192387. 0.0438063
\(455\) 777096. 0.175973
\(456\) −525619. −0.118375
\(457\) −1.53312e6 −0.343388 −0.171694 0.985150i \(-0.554924\pi\)
−0.171694 + 0.985150i \(0.554924\pi\)
\(458\) 33893.4 0.00755008
\(459\) −849187. −0.188136
\(460\) 113397. 0.0249866
\(461\) 2.69249e6 0.590067 0.295033 0.955487i \(-0.404669\pi\)
0.295033 + 0.955487i \(0.404669\pi\)
\(462\) 28346.9 0.00617875
\(463\) −2.40113e6 −0.520550 −0.260275 0.965535i \(-0.583813\pi\)
−0.260275 + 0.965535i \(0.583813\pi\)
\(464\) −6.59748e6 −1.42260
\(465\) 139663. 0.0299535
\(466\) −297214. −0.0634023
\(467\) −7.70828e6 −1.63556 −0.817778 0.575534i \(-0.804794\pi\)
−0.817778 + 0.575534i \(0.804794\pi\)
\(468\) 1.21320e6 0.256047
\(469\) 1.28654e6 0.270079
\(470\) −20136.1 −0.00420465
\(471\) −125533. −0.0260739
\(472\) −111325. −0.0230006
\(473\) −342974. −0.0704869
\(474\) −404754. −0.0827457
\(475\) −5.34623e6 −1.08721
\(476\) −4.33559e6 −0.877064
\(477\) −1.35074e6 −0.271817
\(478\) −22984.7 −0.00460118
\(479\) 4.29567e6 0.855445 0.427723 0.903910i \(-0.359316\pi\)
0.427723 + 0.903910i \(0.359316\pi\)
\(480\) −192846. −0.0382038
\(481\) 3.65550e6 0.720416
\(482\) 787254. 0.154347
\(483\) −268214. −0.0523135
\(484\) 5.02205e6 0.974467
\(485\) 194949. 0.0376328
\(486\) 29623.3 0.00568910
\(487\) −9.17052e6 −1.75215 −0.876076 0.482173i \(-0.839848\pi\)
−0.876076 + 0.482173i \(0.839848\pi\)
\(488\) 1.18647e6 0.225532
\(489\) −1.07904e6 −0.204064
\(490\) −21592.5 −0.00406268
\(491\) −1.90208e6 −0.356062 −0.178031 0.984025i \(-0.556973\pi\)
−0.178031 + 0.984025i \(0.556973\pi\)
\(492\) −1.66437e6 −0.309982
\(493\) −7.68592e6 −1.42423
\(494\) 432204. 0.0796840
\(495\) −60949.3 −0.0111804
\(496\) 1.10434e6 0.201556
\(497\) −1.81465e6 −0.329536
\(498\) 264988. 0.0478799
\(499\) −5.54825e6 −0.997482 −0.498741 0.866751i \(-0.666204\pi\)
−0.498741 + 0.866751i \(0.666204\pi\)
\(500\) −2.69995e6 −0.482982
\(501\) 62154.9 0.0110632
\(502\) 18338.1 0.00324784
\(503\) 5.18973e6 0.914587 0.457294 0.889316i \(-0.348819\pi\)
0.457294 + 0.889316i \(0.348819\pi\)
\(504\) 303688. 0.0532538
\(505\) −1.43223e6 −0.249910
\(506\) −6829.65 −0.00118583
\(507\) 1.33855e6 0.231268
\(508\) 1.92750e6 0.331387
\(509\) 2.94232e6 0.503379 0.251689 0.967808i \(-0.419014\pi\)
0.251689 + 0.967808i \(0.419014\pi\)
\(510\) −73898.2 −0.0125808
\(511\) −4.94391e6 −0.837565
\(512\) −2.54815e6 −0.429586
\(513\) −1.33127e6 −0.223343
\(514\) 412340. 0.0688411
\(515\) 2.47475e6 0.411162
\(516\) −1.82993e6 −0.302560
\(517\) −152985. −0.0251723
\(518\) 455713. 0.0746220
\(519\) 3.55219e6 0.578866
\(520\) 211989. 0.0343799
\(521\) −3.27821e6 −0.529106 −0.264553 0.964371i \(-0.585224\pi\)
−0.264553 + 0.964371i \(0.585224\pi\)
\(522\) 268118. 0.0430676
\(523\) −890700. −0.142389 −0.0711946 0.997462i \(-0.522681\pi\)
−0.0711946 + 0.997462i \(0.522681\pi\)
\(524\) −6.07332e6 −0.966268
\(525\) 3.08890e6 0.489109
\(526\) 121712. 0.0191809
\(527\) 1.28653e6 0.201787
\(528\) −481936. −0.0752324
\(529\) −6.37172e6 −0.989960
\(530\) −117545. −0.0181767
\(531\) −281961. −0.0433963
\(532\) −6.79691e6 −1.04120
\(533\) 2.74799e6 0.418983
\(534\) 278815. 0.0423120
\(535\) −1.97656e6 −0.298556
\(536\) 350963. 0.0527654
\(537\) −6.55980e6 −0.981646
\(538\) 535016. 0.0796913
\(539\) −164050. −0.0243223
\(540\) −325194. −0.0479908
\(541\) 5.79897e6 0.851839 0.425920 0.904761i \(-0.359951\pi\)
0.425920 + 0.904761i \(0.359951\pi\)
\(542\) −101593. −0.0148547
\(543\) 3.68130e6 0.535799
\(544\) −1.77643e6 −0.257366
\(545\) 1.22345e6 0.176439
\(546\) −249715. −0.0358478
\(547\) 2.82959e6 0.404348 0.202174 0.979350i \(-0.435199\pi\)
0.202174 + 0.979350i \(0.435199\pi\)
\(548\) −8.33753e6 −1.18600
\(549\) 3.00506e6 0.425522
\(550\) 78653.8 0.0110870
\(551\) −1.20492e7 −1.69075
\(552\) −73167.8 −0.0102205
\(553\) −1.05094e7 −1.46139
\(554\) −632622. −0.0875729
\(555\) −979839. −0.135028
\(556\) 4.74782e6 0.651339
\(557\) −9.20725e6 −1.25745 −0.628726 0.777627i \(-0.716423\pi\)
−0.628726 + 0.777627i \(0.716423\pi\)
\(558\) −44879.7 −0.00610189
\(559\) 3.02134e6 0.408950
\(560\) −1.64704e6 −0.221939
\(561\) −561445. −0.0753183
\(562\) −969964. −0.129543
\(563\) 4.99478e6 0.664118 0.332059 0.943259i \(-0.392257\pi\)
0.332059 + 0.943259i \(0.392257\pi\)
\(564\) −816248. −0.108050
\(565\) 3.61504e6 0.476423
\(566\) 252089. 0.0330759
\(567\) 769170. 0.100476
\(568\) −495030. −0.0643814
\(569\) 1.02729e6 0.133018 0.0665092 0.997786i \(-0.478814\pi\)
0.0665092 + 0.997786i \(0.478814\pi\)
\(570\) −115850. −0.0149352
\(571\) 7.08898e6 0.909900 0.454950 0.890517i \(-0.349657\pi\)
0.454950 + 0.890517i \(0.349657\pi\)
\(572\) 802118. 0.102506
\(573\) 4.94933e6 0.629738
\(574\) 342578. 0.0433990
\(575\) −744211. −0.0938700
\(576\) −2.52978e6 −0.317706
\(577\) −4.15153e6 −0.519121 −0.259561 0.965727i \(-0.583578\pi\)
−0.259561 + 0.965727i \(0.583578\pi\)
\(578\) 31578.6 0.00393163
\(579\) −2.45275e6 −0.304058
\(580\) −2.94331e6 −0.363300
\(581\) 6.88042e6 0.845618
\(582\) −62645.5 −0.00766624
\(583\) −893054. −0.108819
\(584\) −1.34868e6 −0.163635
\(585\) 536918. 0.0648661
\(586\) 43457.5 0.00522782
\(587\) 1.65158e7 1.97835 0.989176 0.146737i \(-0.0468771\pi\)
0.989176 + 0.146737i \(0.0468771\pi\)
\(588\) −875287. −0.104402
\(589\) 2.01689e6 0.239549
\(590\) −24536.9 −0.00290195
\(591\) 1.77255e6 0.208752
\(592\) −7.74775e6 −0.908597
\(593\) 7.04831e6 0.823092 0.411546 0.911389i \(-0.364989\pi\)
0.411546 + 0.911389i \(0.364989\pi\)
\(594\) 19585.7 0.00227757
\(595\) −1.91877e6 −0.222193
\(596\) 1.13331e7 1.30687
\(597\) 5.16566e6 0.593184
\(598\) 60164.0 0.00687993
\(599\) 1.26382e7 1.43919 0.719594 0.694395i \(-0.244328\pi\)
0.719594 + 0.694395i \(0.244328\pi\)
\(600\) 842639. 0.0955572
\(601\) 6.61706e6 0.747272 0.373636 0.927575i \(-0.378111\pi\)
0.373636 + 0.927575i \(0.378111\pi\)
\(602\) 376656. 0.0423598
\(603\) 888906. 0.0995549
\(604\) 7.51778e6 0.838489
\(605\) 2.22256e6 0.246868
\(606\) 460237. 0.0509096
\(607\) −2.10068e6 −0.231413 −0.115707 0.993283i \(-0.536913\pi\)
−0.115707 + 0.993283i \(0.536913\pi\)
\(608\) −2.78492e6 −0.305529
\(609\) 6.96169e6 0.760627
\(610\) 261507. 0.0284550
\(611\) 1.34768e6 0.146044
\(612\) −2.99558e6 −0.323298
\(613\) 1.51680e7 1.63034 0.815168 0.579225i \(-0.196645\pi\)
0.815168 + 0.579225i \(0.196645\pi\)
\(614\) −316391. −0.0338691
\(615\) −736585. −0.0785299
\(616\) 200785. 0.0213196
\(617\) −1.74862e6 −0.184920 −0.0924600 0.995716i \(-0.529473\pi\)
−0.0924600 + 0.995716i \(0.529473\pi\)
\(618\) −795244. −0.0837585
\(619\) −1.06999e7 −1.12241 −0.561206 0.827676i \(-0.689662\pi\)
−0.561206 + 0.827676i \(0.689662\pi\)
\(620\) 492673. 0.0514730
\(621\) −185317. −0.0192835
\(622\) 415810. 0.0430942
\(623\) 7.23944e6 0.747282
\(624\) 4.24550e6 0.436483
\(625\) 7.95380e6 0.814469
\(626\) −824939. −0.0841368
\(627\) −880179. −0.0894133
\(628\) −442829. −0.0448061
\(629\) −9.02597e6 −0.909635
\(630\) 66934.9 0.00671895
\(631\) 1.06482e7 1.06464 0.532322 0.846542i \(-0.321319\pi\)
0.532322 + 0.846542i \(0.321319\pi\)
\(632\) −2.86693e6 −0.285512
\(633\) 7.31657e6 0.725769
\(634\) −1.33761e6 −0.132162
\(635\) 853038. 0.0839526
\(636\) −4.76488e6 −0.467099
\(637\) 1.44516e6 0.141113
\(638\) 177268. 0.0172417
\(639\) −1.25379e6 −0.121471
\(640\) −905821. −0.0874163
\(641\) 1.71884e7 1.65231 0.826155 0.563443i \(-0.190523\pi\)
0.826155 + 0.563443i \(0.190523\pi\)
\(642\) 635155. 0.0608194
\(643\) 9.90174e6 0.944461 0.472230 0.881475i \(-0.343449\pi\)
0.472230 + 0.881475i \(0.343449\pi\)
\(644\) −946151. −0.0898971
\(645\) −809858. −0.0766495
\(646\) −1.06718e6 −0.100613
\(647\) 4.53741e6 0.426136 0.213068 0.977037i \(-0.431654\pi\)
0.213068 + 0.977037i \(0.431654\pi\)
\(648\) 209826. 0.0196301
\(649\) −186420. −0.0173733
\(650\) −692881. −0.0643243
\(651\) −1.16530e6 −0.107767
\(652\) −3.80643e6 −0.350670
\(653\) −5.46204e6 −0.501270 −0.250635 0.968082i \(-0.580639\pi\)
−0.250635 + 0.968082i \(0.580639\pi\)
\(654\) −393148. −0.0359428
\(655\) −2.68782e6 −0.244791
\(656\) −5.82430e6 −0.528426
\(657\) −3.41589e6 −0.308738
\(658\) 168009. 0.0151275
\(659\) 1.36246e6 0.122211 0.0611056 0.998131i \(-0.480537\pi\)
0.0611056 + 0.998131i \(0.480537\pi\)
\(660\) −215004. −0.0192126
\(661\) 1.11826e7 0.995497 0.497748 0.867321i \(-0.334160\pi\)
0.497748 + 0.867321i \(0.334160\pi\)
\(662\) −546944. −0.0485062
\(663\) 4.94591e6 0.436981
\(664\) 1.87695e6 0.165209
\(665\) −3.00805e6 −0.263773
\(666\) 314865. 0.0275067
\(667\) −1.67729e6 −0.145980
\(668\) 219257. 0.0190113
\(669\) 1.90247e6 0.164344
\(670\) 77354.7 0.00665732
\(671\) 1.98682e6 0.170354
\(672\) 1.60904e6 0.137450
\(673\) −2.22259e7 −1.89157 −0.945785 0.324794i \(-0.894705\pi\)
−0.945785 + 0.324794i \(0.894705\pi\)
\(674\) 687850. 0.0583235
\(675\) 2.13421e6 0.180292
\(676\) 4.72187e6 0.397418
\(677\) −7.09861e6 −0.595253 −0.297627 0.954682i \(-0.596195\pi\)
−0.297627 + 0.954682i \(0.596195\pi\)
\(678\) −1.16167e6 −0.0970529
\(679\) −1.62659e6 −0.135395
\(680\) −523432. −0.0434098
\(681\) 3.45141e6 0.285187
\(682\) −29672.5 −0.00244283
\(683\) −2.04470e7 −1.67717 −0.838585 0.544771i \(-0.816616\pi\)
−0.838585 + 0.544771i \(0.816616\pi\)
\(684\) −4.69618e6 −0.383799
\(685\) −3.68987e6 −0.300458
\(686\) 1.16863e6 0.0948128
\(687\) 608046. 0.0491524
\(688\) −6.40368e6 −0.515773
\(689\) 7.86713e6 0.631347
\(690\) −16126.7 −0.00128950
\(691\) 7.78175e6 0.619987 0.309993 0.950739i \(-0.399673\pi\)
0.309993 + 0.950739i \(0.399673\pi\)
\(692\) 1.25307e7 0.994740
\(693\) 508542. 0.0402248
\(694\) 1.18052e6 0.0930411
\(695\) 2.10120e6 0.165008
\(696\) 1.89912e6 0.148604
\(697\) −6.78519e6 −0.529029
\(698\) −423579. −0.0329075
\(699\) −5.33201e6 −0.412760
\(700\) 1.08964e7 0.840498
\(701\) −1.01109e7 −0.777136 −0.388568 0.921420i \(-0.627030\pi\)
−0.388568 + 0.921420i \(0.627030\pi\)
\(702\) −172535. −0.0132140
\(703\) −1.41500e7 −1.07986
\(704\) −1.67258e6 −0.127191
\(705\) −361240. −0.0273730
\(706\) −539179. −0.0407119
\(707\) 1.19500e7 0.899127
\(708\) −994643. −0.0745734
\(709\) 5.55179e6 0.414780 0.207390 0.978258i \(-0.433503\pi\)
0.207390 + 0.978258i \(0.433503\pi\)
\(710\) −109108. −0.00812290
\(711\) −7.26127e6 −0.538689
\(712\) 1.97489e6 0.145997
\(713\) 280757. 0.0206827
\(714\) 616583. 0.0452633
\(715\) 354987. 0.0259685
\(716\) −2.31403e7 −1.68689
\(717\) −412344. −0.0299545
\(718\) −979140. −0.0708816
\(719\) 1.36153e7 0.982215 0.491107 0.871099i \(-0.336592\pi\)
0.491107 + 0.871099i \(0.336592\pi\)
\(720\) −1.13799e6 −0.0818099
\(721\) −2.06485e7 −1.47928
\(722\) −430818. −0.0307575
\(723\) 1.41233e7 1.00483
\(724\) 1.29861e7 0.920731
\(725\) 1.93165e7 1.36485
\(726\) −714206. −0.0502901
\(727\) 7.71813e6 0.541597 0.270798 0.962636i \(-0.412712\pi\)
0.270798 + 0.962636i \(0.412712\pi\)
\(728\) −1.76877e6 −0.123692
\(729\) 531441. 0.0370370
\(730\) −297258. −0.0206456
\(731\) −7.46015e6 −0.516362
\(732\) 1.06006e7 0.731230
\(733\) −2.00055e7 −1.37527 −0.687636 0.726056i \(-0.741351\pi\)
−0.687636 + 0.726056i \(0.741351\pi\)
\(734\) −785779. −0.0538345
\(735\) −387368. −0.0264488
\(736\) −387669. −0.0263795
\(737\) 587706. 0.0398558
\(738\) 236697. 0.0159975
\(739\) −1.40919e7 −0.949204 −0.474602 0.880201i \(-0.657408\pi\)
−0.474602 + 0.880201i \(0.657408\pi\)
\(740\) −3.45647e6 −0.232035
\(741\) 7.75371e6 0.518757
\(742\) 980758. 0.0653961
\(743\) 1.01829e7 0.676706 0.338353 0.941019i \(-0.390130\pi\)
0.338353 + 0.941019i \(0.390130\pi\)
\(744\) −317889. −0.0210544
\(745\) 5.01558e6 0.331078
\(746\) −1.43405e6 −0.0943444
\(747\) 4.75387e6 0.311707
\(748\) −1.98055e6 −0.129429
\(749\) 1.64918e7 1.07415
\(750\) 383971. 0.0249256
\(751\) −2.24603e7 −1.45317 −0.726583 0.687079i \(-0.758893\pi\)
−0.726583 + 0.687079i \(0.758893\pi\)
\(752\) −2.85638e6 −0.184193
\(753\) 328984. 0.0211440
\(754\) −1.56160e6 −0.100033
\(755\) 3.32708e6 0.212420
\(756\) 2.71332e6 0.172662
\(757\) −2.42736e7 −1.53955 −0.769777 0.638313i \(-0.779633\pi\)
−0.769777 + 0.638313i \(0.779633\pi\)
\(758\) −833471. −0.0526887
\(759\) −122523. −0.00771996
\(760\) −820584. −0.0515334
\(761\) 2.19171e7 1.37190 0.685948 0.727651i \(-0.259388\pi\)
0.685948 + 0.727651i \(0.259388\pi\)
\(762\) −274118. −0.0171021
\(763\) −1.02081e7 −0.634795
\(764\) 1.74592e7 1.08216
\(765\) −1.32573e6 −0.0819033
\(766\) 2.46535e6 0.151812
\(767\) 1.64222e6 0.100796
\(768\) −8.70368e6 −0.532476
\(769\) 1.95692e7 1.19332 0.596660 0.802494i \(-0.296494\pi\)
0.596660 + 0.802494i \(0.296494\pi\)
\(770\) 44254.5 0.00268986
\(771\) 7.39736e6 0.448168
\(772\) −8.65229e6 −0.522502
\(773\) 2.41099e7 1.45127 0.725634 0.688081i \(-0.241547\pi\)
0.725634 + 0.688081i \(0.241547\pi\)
\(774\) 260243. 0.0156144
\(775\) −3.23335e6 −0.193374
\(776\) −443728. −0.0264522
\(777\) 8.17547e6 0.485803
\(778\) −2.02824e6 −0.120135
\(779\) −1.06371e7 −0.628031
\(780\) 1.89403e6 0.111468
\(781\) −828955. −0.0486299
\(782\) −148554. −0.00868695
\(783\) 4.81003e6 0.280378
\(784\) −3.06298e6 −0.177973
\(785\) −195979. −0.0113510
\(786\) 863712. 0.0498669
\(787\) −3.33270e7 −1.91805 −0.959025 0.283321i \(-0.908564\pi\)
−0.959025 + 0.283321i \(0.908564\pi\)
\(788\) 6.25284e6 0.358725
\(789\) 2.18351e6 0.124871
\(790\) −631892. −0.0360226
\(791\) −3.01628e7 −1.71408
\(792\) 138728. 0.00785872
\(793\) −1.75024e7 −0.988356
\(794\) −1.16184e6 −0.0654026
\(795\) −2.10875e6 −0.118333
\(796\) 1.82223e7 1.01935
\(797\) −864232. −0.0481930 −0.0240965 0.999710i \(-0.507671\pi\)
−0.0240965 + 0.999710i \(0.507671\pi\)
\(798\) 966618. 0.0537338
\(799\) −3.32763e6 −0.184403
\(800\) 4.46460e6 0.246636
\(801\) 5.00193e6 0.275459
\(802\) 1.62545e6 0.0892358
\(803\) −2.25844e6 −0.123600
\(804\) 3.13570e6 0.171078
\(805\) −418730. −0.0227742
\(806\) 261392. 0.0141728
\(807\) 9.59815e6 0.518805
\(808\) 3.25993e6 0.175663
\(809\) 1.13424e7 0.609303 0.304651 0.952464i \(-0.401460\pi\)
0.304651 + 0.952464i \(0.401460\pi\)
\(810\) 46247.2 0.00247670
\(811\) 3.43446e6 0.183361 0.0916805 0.995788i \(-0.470776\pi\)
0.0916805 + 0.995788i \(0.470776\pi\)
\(812\) 2.45580e7 1.30708
\(813\) −1.82256e6 −0.0967067
\(814\) 208175. 0.0110120
\(815\) −1.68458e6 −0.0888376
\(816\) −1.04828e7 −0.551125
\(817\) −1.16953e7 −0.612993
\(818\) −1.15426e6 −0.0603142
\(819\) −4.47987e6 −0.233376
\(820\) −2.59837e6 −0.134948
\(821\) 1.81504e7 0.939782 0.469891 0.882724i \(-0.344293\pi\)
0.469891 + 0.882724i \(0.344293\pi\)
\(822\) 1.18572e6 0.0612070
\(823\) −1.63665e6 −0.0842281 −0.0421141 0.999113i \(-0.513409\pi\)
−0.0421141 + 0.999113i \(0.513409\pi\)
\(824\) −5.63283e6 −0.289007
\(825\) 1.41105e6 0.0721782
\(826\) 204728. 0.0104406
\(827\) −1.28654e7 −0.654125 −0.327063 0.945003i \(-0.606059\pi\)
−0.327063 + 0.945003i \(0.606059\pi\)
\(828\) −653722. −0.0331373
\(829\) −8.03655e6 −0.406147 −0.203073 0.979164i \(-0.565093\pi\)
−0.203073 + 0.979164i \(0.565093\pi\)
\(830\) 413693. 0.0208441
\(831\) −1.13492e7 −0.570116
\(832\) 1.47342e7 0.737933
\(833\) −3.56831e6 −0.178176
\(834\) −675207. −0.0336141
\(835\) 97034.7 0.00481627
\(836\) −3.10491e6 −0.153650
\(837\) −805139. −0.0397244
\(838\) −2.36738e6 −0.116455
\(839\) −3.59741e7 −1.76435 −0.882176 0.470920i \(-0.843922\pi\)
−0.882176 + 0.470920i \(0.843922\pi\)
\(840\) 474110. 0.0231836
\(841\) 2.30241e7 1.12251
\(842\) −798566. −0.0388178
\(843\) −1.74011e7 −0.843350
\(844\) 2.58099e7 1.24718
\(845\) 2.08972e6 0.100681
\(846\) 116082. 0.00557621
\(847\) −1.85444e7 −0.888185
\(848\) −1.66742e7 −0.796262
\(849\) 4.52246e6 0.215330
\(850\) 1.71083e6 0.0812192
\(851\) −1.96972e6 −0.0932355
\(852\) −4.42287e6 −0.208740
\(853\) −2.61300e7 −1.22961 −0.614803 0.788680i \(-0.710765\pi\)
−0.614803 + 0.788680i \(0.710765\pi\)
\(854\) −2.18193e6 −0.102376
\(855\) −2.07835e6 −0.0972306
\(856\) 4.49889e6 0.209856
\(857\) −2.58354e7 −1.20161 −0.600804 0.799396i \(-0.705153\pi\)
−0.600804 + 0.799396i \(0.705153\pi\)
\(858\) −114073. −0.00529009
\(859\) −2.85214e7 −1.31883 −0.659415 0.751779i \(-0.729196\pi\)
−0.659415 + 0.751779i \(0.729196\pi\)
\(860\) −2.85685e6 −0.131717
\(861\) 6.14583e6 0.282535
\(862\) 1.92732e6 0.0883460
\(863\) −2.80905e7 −1.28390 −0.641952 0.766745i \(-0.721875\pi\)
−0.641952 + 0.766745i \(0.721875\pi\)
\(864\) 1.11173e6 0.0506660
\(865\) 5.54560e6 0.252004
\(866\) 48973.4 0.00221904
\(867\) 566518. 0.0255956
\(868\) −4.11070e6 −0.185190
\(869\) −4.80083e6 −0.215659
\(870\) 418580. 0.0187491
\(871\) −5.17725e6 −0.231235
\(872\) −2.78473e6 −0.124020
\(873\) −1.12386e6 −0.0499086
\(874\) −232888. −0.0103126
\(875\) 9.96981e6 0.440217
\(876\) −1.20499e7 −0.530544
\(877\) 1.23614e7 0.542712 0.271356 0.962479i \(-0.412528\pi\)
0.271356 + 0.962479i \(0.412528\pi\)
\(878\) −2.22263e6 −0.0973041
\(879\) 779626. 0.0340341
\(880\) −752387. −0.0327518
\(881\) −1.44013e7 −0.625119 −0.312560 0.949898i \(-0.601186\pi\)
−0.312560 + 0.949898i \(0.601186\pi\)
\(882\) 124478. 0.00538793
\(883\) 76484.3 0.00330119 0.00165059 0.999999i \(-0.499475\pi\)
0.00165059 + 0.999999i \(0.499475\pi\)
\(884\) 1.74472e7 0.750921
\(885\) −440191. −0.0188922
\(886\) 3.08178e6 0.131891
\(887\) −4.12317e7 −1.75963 −0.879817 0.475312i \(-0.842335\pi\)
−0.879817 + 0.475312i \(0.842335\pi\)
\(888\) 2.23024e6 0.0949114
\(889\) −7.11748e6 −0.302045
\(890\) 435280. 0.0184202
\(891\) 351366. 0.0148274
\(892\) 6.71115e6 0.282413
\(893\) −5.21672e6 −0.218912
\(894\) −1.61172e6 −0.0674445
\(895\) −1.02410e7 −0.427351
\(896\) 7.55788e6 0.314507
\(897\) 1.07934e6 0.0447896
\(898\) −2.70641e6 −0.111996
\(899\) −7.28725e6 −0.300721
\(900\) 7.52861e6 0.309819
\(901\) −1.94251e7 −0.797172
\(902\) 156494. 0.00640443
\(903\) 6.75720e6 0.275770
\(904\) −8.22828e6 −0.334879
\(905\) 5.74715e6 0.233255
\(906\) −1.06913e6 −0.0432725
\(907\) 3.00251e7 1.21190 0.605950 0.795503i \(-0.292793\pi\)
0.605950 + 0.795503i \(0.292793\pi\)
\(908\) 1.21752e7 0.490073
\(909\) 8.25663e6 0.331431
\(910\) −389849. −0.0156060
\(911\) −7.61632e6 −0.304053 −0.152026 0.988376i \(-0.548580\pi\)
−0.152026 + 0.988376i \(0.548580\pi\)
\(912\) −1.64338e7 −0.654262
\(913\) 3.14306e6 0.124789
\(914\) 769124. 0.0304531
\(915\) 4.69143e6 0.185247
\(916\) 2.14494e6 0.0844648
\(917\) 2.24263e7 0.880712
\(918\) 426015. 0.0166847
\(919\) 1.41234e7 0.551634 0.275817 0.961210i \(-0.411052\pi\)
0.275817 + 0.961210i \(0.411052\pi\)
\(920\) −114228. −0.00444941
\(921\) −5.67604e6 −0.220494
\(922\) −1.35075e6 −0.0523296
\(923\) 7.30247e6 0.282140
\(924\) 1.79393e6 0.0691234
\(925\) 2.26844e7 0.871711
\(926\) 1.20458e6 0.0461646
\(927\) −1.42666e7 −0.545283
\(928\) 1.00622e7 0.383551
\(929\) −2.95615e7 −1.12380 −0.561898 0.827206i \(-0.689929\pi\)
−0.561898 + 0.827206i \(0.689929\pi\)
\(930\) −70065.0 −0.00265640
\(931\) −5.59405e6 −0.211520
\(932\) −1.88091e7 −0.709299
\(933\) 7.45961e6 0.280551
\(934\) 3.86704e6 0.145048
\(935\) −876515. −0.0327892
\(936\) −1.22209e6 −0.0455946
\(937\) 3.40344e7 1.26639 0.633197 0.773990i \(-0.281742\pi\)
0.633197 + 0.773990i \(0.281742\pi\)
\(938\) −645423. −0.0239518
\(939\) −1.47994e7 −0.547746
\(940\) −1.27431e6 −0.0470386
\(941\) 2.51116e7 0.924485 0.462242 0.886754i \(-0.347045\pi\)
0.462242 + 0.886754i \(0.347045\pi\)
\(942\) 62976.6 0.00231234
\(943\) −1.48072e6 −0.0542243
\(944\) −3.48066e6 −0.127125
\(945\) 1.20081e6 0.0437416
\(946\) 172061. 0.00625108
\(947\) −2.02733e7 −0.734599 −0.367300 0.930103i \(-0.619718\pi\)
−0.367300 + 0.930103i \(0.619718\pi\)
\(948\) −2.56148e7 −0.925699
\(949\) 1.98951e7 0.717102
\(950\) 2.68206e6 0.0964184
\(951\) −2.39967e7 −0.860398
\(952\) 4.36735e6 0.156180
\(953\) 1.24755e7 0.444965 0.222483 0.974937i \(-0.428584\pi\)
0.222483 + 0.974937i \(0.428584\pi\)
\(954\) 677633. 0.0241059
\(955\) 7.72678e6 0.274151
\(956\) −1.45458e6 −0.0514746
\(957\) 3.18018e6 0.112246
\(958\) −2.15503e6 −0.0758645
\(959\) 3.07871e7 1.08099
\(960\) −3.94943e6 −0.138311
\(961\) −2.74094e7 −0.957393
\(962\) −1.83387e6 −0.0638896
\(963\) 1.13946e7 0.395945
\(964\) 4.98212e7 1.72672
\(965\) −3.82917e6 −0.132369
\(966\) 134556. 0.00463939
\(967\) 3.79545e7 1.30526 0.652630 0.757677i \(-0.273666\pi\)
0.652630 + 0.757677i \(0.273666\pi\)
\(968\) −5.05883e6 −0.173525
\(969\) −1.91451e7 −0.655009
\(970\) −97800.7 −0.00333743
\(971\) −5.56592e6 −0.189447 −0.0947237 0.995504i \(-0.530197\pi\)
−0.0947237 + 0.995504i \(0.530197\pi\)
\(972\) 1.87471e6 0.0636455
\(973\) −1.75318e7 −0.593667
\(974\) 4.60061e6 0.155388
\(975\) −1.24302e7 −0.418763
\(976\) 3.70959e7 1.24653
\(977\) −4.95886e7 −1.66206 −0.831028 0.556230i \(-0.812247\pi\)
−0.831028 + 0.556230i \(0.812247\pi\)
\(978\) 541328. 0.0180973
\(979\) 3.30706e6 0.110277
\(980\) −1.36648e6 −0.0454503
\(981\) −7.05306e6 −0.233994
\(982\) 954224. 0.0315771
\(983\) 1.16536e7 0.384660 0.192330 0.981330i \(-0.438396\pi\)
0.192330 + 0.981330i \(0.438396\pi\)
\(984\) 1.67656e6 0.0551990
\(985\) 2.76726e6 0.0908783
\(986\) 3.85582e6 0.126306
\(987\) 3.01407e6 0.0984829
\(988\) 2.73519e7 0.891447
\(989\) −1.62802e6 −0.0529259
\(990\) 30576.7 0.000991521 0
\(991\) −1.62828e7 −0.526679 −0.263340 0.964703i \(-0.584824\pi\)
−0.263340 + 0.964703i \(0.584824\pi\)
\(992\) −1.68429e6 −0.0543422
\(993\) −9.81214e6 −0.315784
\(994\) 910363. 0.0292246
\(995\) 8.06450e6 0.258238
\(996\) 1.67697e7 0.535646
\(997\) 1.47322e7 0.469386 0.234693 0.972070i \(-0.424592\pi\)
0.234693 + 0.972070i \(0.424592\pi\)
\(998\) 2.78341e6 0.0884609
\(999\) 5.64866e6 0.179074
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 177.6.a.b.1.7 12
3.2 odd 2 531.6.a.d.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.7 12 1.1 even 1 trivial
531.6.a.d.1.6 12 3.2 odd 2