Properties

Label 21.4.a.c.1.1
Level $21$
Weight $4$
Character 21.1
Self dual yes
Analytic conductor $1.239$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 21.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.27492 q^{2} +3.00000 q^{3} +19.8248 q^{4} +10.5498 q^{5} -15.8248 q^{6} +7.00000 q^{7} -62.3746 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.27492 q^{2} +3.00000 q^{3} +19.8248 q^{4} +10.5498 q^{5} -15.8248 q^{6} +7.00000 q^{7} -62.3746 q^{8} +9.00000 q^{9} -55.6495 q^{10} +34.7492 q^{11} +59.4743 q^{12} -37.2990 q^{13} -36.9244 q^{14} +31.6495 q^{15} +170.423 q^{16} -10.5498 q^{17} -47.4743 q^{18} -58.5980 q^{19} +209.148 q^{20} +21.0000 q^{21} -183.299 q^{22} -125.347 q^{23} -187.124 q^{24} -13.7010 q^{25} +196.749 q^{26} +27.0000 q^{27} +138.773 q^{28} -35.4020 q^{29} -166.949 q^{30} +291.794 q^{31} -399.969 q^{32} +104.248 q^{33} +55.6495 q^{34} +73.8488 q^{35} +178.423 q^{36} -259.897 q^{37} +309.100 q^{38} -111.897 q^{39} -658.042 q^{40} -338.248 q^{41} -110.773 q^{42} +6.80397 q^{43} +688.894 q^{44} +94.9485 q^{45} +661.196 q^{46} +250.694 q^{47} +511.268 q^{48} +49.0000 q^{49} +72.2716 q^{50} -31.6495 q^{51} -739.444 q^{52} -536.900 q^{53} -142.423 q^{54} +366.598 q^{55} -436.622 q^{56} -175.794 q^{57} +186.743 q^{58} -35.8904 q^{59} +627.444 q^{60} +57.7940 q^{61} -1539.19 q^{62} +63.0000 q^{63} +746.423 q^{64} -393.498 q^{65} -549.897 q^{66} +481.691 q^{67} -209.148 q^{68} -376.042 q^{69} -389.547 q^{70} +363.752 q^{71} -561.371 q^{72} +581.299 q^{73} +1370.94 q^{74} -41.1030 q^{75} -1161.69 q^{76} +243.244 q^{77} +590.248 q^{78} -693.691 q^{79} +1797.93 q^{80} +81.0000 q^{81} +1784.23 q^{82} +1334.39 q^{83} +416.320 q^{84} -111.299 q^{85} -35.8904 q^{86} -106.206 q^{87} -2167.47 q^{88} -353.038 q^{89} -500.846 q^{90} -261.093 q^{91} -2484.98 q^{92} +875.382 q^{93} -1322.39 q^{94} -618.199 q^{95} -1199.91 q^{96} +1445.88 q^{97} -258.471 q^{98} +312.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 6q^{3} + 17q^{4} + 6q^{5} - 9q^{6} + 14q^{7} - 87q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + 6q^{3} + 17q^{4} + 6q^{5} - 9q^{6} + 14q^{7} - 87q^{8} + 18q^{9} - 66q^{10} - 6q^{11} + 51q^{12} + 16q^{13} - 21q^{14} + 18q^{15} + 137q^{16} - 6q^{17} - 27q^{18} + 64q^{19} + 222q^{20} + 42q^{21} - 276q^{22} + 6q^{23} - 261q^{24} - 118q^{25} + 318q^{26} + 54q^{27} + 119q^{28} - 252q^{29} - 198q^{30} + 40q^{31} - 279q^{32} - 18q^{33} + 66q^{34} + 42q^{35} + 153q^{36} - 248q^{37} + 588q^{38} + 48q^{39} - 546q^{40} - 450q^{41} - 63q^{42} + 376q^{43} + 804q^{44} + 54q^{45} + 960q^{46} - 12q^{47} + 411q^{48} + 98q^{49} - 165q^{50} - 18q^{51} - 890q^{52} - 1104q^{53} - 81q^{54} + 552q^{55} - 609q^{56} + 192q^{57} - 306q^{58} + 804q^{59} + 666q^{60} - 428q^{61} - 2112q^{62} + 126q^{63} + 1289q^{64} - 636q^{65} - 828q^{66} + 148q^{67} - 222q^{68} + 18q^{69} - 462q^{70} + 954q^{71} - 783q^{72} + 1072q^{73} + 1398q^{74} - 354q^{75} - 1508q^{76} - 42q^{77} + 954q^{78} - 572q^{79} + 1950q^{80} + 162q^{81} + 1530q^{82} + 1944q^{83} + 357q^{84} - 132q^{85} + 804q^{86} - 756q^{87} - 1164q^{88} + 366q^{89} - 594q^{90} + 112q^{91} - 2856q^{92} + 120q^{93} - 1920q^{94} - 1176q^{95} - 837q^{96} + 808q^{97} - 147q^{98} - 54q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.27492 −1.86496 −0.932482 0.361215i \(-0.882362\pi\)
−0.932482 + 0.361215i \(0.882362\pi\)
\(3\) 3.00000 0.577350
\(4\) 19.8248 2.47809
\(5\) 10.5498 0.943606 0.471803 0.881704i \(-0.343603\pi\)
0.471803 + 0.881704i \(0.343603\pi\)
\(6\) −15.8248 −1.07674
\(7\) 7.00000 0.377964
\(8\) −62.3746 −2.75659
\(9\) 9.00000 0.333333
\(10\) −55.6495 −1.75979
\(11\) 34.7492 0.952479 0.476240 0.879316i \(-0.342000\pi\)
0.476240 + 0.879316i \(0.342000\pi\)
\(12\) 59.4743 1.43073
\(13\) −37.2990 −0.795760 −0.397880 0.917437i \(-0.630254\pi\)
−0.397880 + 0.917437i \(0.630254\pi\)
\(14\) −36.9244 −0.704890
\(15\) 31.6495 0.544791
\(16\) 170.423 2.66286
\(17\) −10.5498 −0.150512 −0.0752562 0.997164i \(-0.523977\pi\)
−0.0752562 + 0.997164i \(0.523977\pi\)
\(18\) −47.4743 −0.621655
\(19\) −58.5980 −0.707542 −0.353771 0.935332i \(-0.615101\pi\)
−0.353771 + 0.935332i \(0.615101\pi\)
\(20\) 209.148 2.33834
\(21\) 21.0000 0.218218
\(22\) −183.299 −1.77634
\(23\) −125.347 −1.13638 −0.568189 0.822898i \(-0.692356\pi\)
−0.568189 + 0.822898i \(0.692356\pi\)
\(24\) −187.124 −1.59152
\(25\) −13.7010 −0.109608
\(26\) 196.749 1.48406
\(27\) 27.0000 0.192450
\(28\) 138.773 0.936631
\(29\) −35.4020 −0.226689 −0.113345 0.993556i \(-0.536156\pi\)
−0.113345 + 0.993556i \(0.536156\pi\)
\(30\) −166.949 −1.01602
\(31\) 291.794 1.69057 0.845286 0.534313i \(-0.179430\pi\)
0.845286 + 0.534313i \(0.179430\pi\)
\(32\) −399.969 −2.20954
\(33\) 104.248 0.549914
\(34\) 55.6495 0.280700
\(35\) 73.8488 0.356649
\(36\) 178.423 0.826031
\(37\) −259.897 −1.15478 −0.577389 0.816469i \(-0.695928\pi\)
−0.577389 + 0.816469i \(0.695928\pi\)
\(38\) 309.100 1.31954
\(39\) −111.897 −0.459432
\(40\) −658.042 −2.60114
\(41\) −338.248 −1.28842 −0.644212 0.764847i \(-0.722815\pi\)
−0.644212 + 0.764847i \(0.722815\pi\)
\(42\) −110.773 −0.406969
\(43\) 6.80397 0.0241301 0.0120651 0.999927i \(-0.496159\pi\)
0.0120651 + 0.999927i \(0.496159\pi\)
\(44\) 688.894 2.36033
\(45\) 94.9485 0.314535
\(46\) 661.196 2.11931
\(47\) 250.694 0.778033 0.389016 0.921231i \(-0.372815\pi\)
0.389016 + 0.921231i \(0.372815\pi\)
\(48\) 511.268 1.53740
\(49\) 49.0000 0.142857
\(50\) 72.2716 0.204415
\(51\) −31.6495 −0.0868984
\(52\) −739.444 −1.97197
\(53\) −536.900 −1.39149 −0.695745 0.718289i \(-0.744925\pi\)
−0.695745 + 0.718289i \(0.744925\pi\)
\(54\) −142.423 −0.358913
\(55\) 366.598 0.898765
\(56\) −436.622 −1.04189
\(57\) −175.794 −0.408500
\(58\) 186.743 0.422767
\(59\) −35.8904 −0.0791955 −0.0395977 0.999216i \(-0.512608\pi\)
−0.0395977 + 0.999216i \(0.512608\pi\)
\(60\) 627.444 1.35004
\(61\) 57.7940 0.121308 0.0606538 0.998159i \(-0.480681\pi\)
0.0606538 + 0.998159i \(0.480681\pi\)
\(62\) −1539.19 −3.15286
\(63\) 63.0000 0.125988
\(64\) 746.423 1.45786
\(65\) −393.498 −0.750884
\(66\) −549.897 −1.02557
\(67\) 481.691 0.878327 0.439164 0.898407i \(-0.355275\pi\)
0.439164 + 0.898407i \(0.355275\pi\)
\(68\) −209.148 −0.372984
\(69\) −376.042 −0.656088
\(70\) −389.547 −0.665139
\(71\) 363.752 0.608021 0.304010 0.952669i \(-0.401674\pi\)
0.304010 + 0.952669i \(0.401674\pi\)
\(72\) −561.371 −0.918864
\(73\) 581.299 0.931999 0.465999 0.884785i \(-0.345695\pi\)
0.465999 + 0.884785i \(0.345695\pi\)
\(74\) 1370.94 2.15362
\(75\) −41.1030 −0.0632822
\(76\) −1161.69 −1.75336
\(77\) 243.244 0.360003
\(78\) 590.248 0.856825
\(79\) −693.691 −0.987928 −0.493964 0.869482i \(-0.664453\pi\)
−0.493964 + 0.869482i \(0.664453\pi\)
\(80\) 1797.93 2.51269
\(81\) 81.0000 0.111111
\(82\) 1784.23 2.40287
\(83\) 1334.39 1.76468 0.882341 0.470611i \(-0.155967\pi\)
0.882341 + 0.470611i \(0.155967\pi\)
\(84\) 416.320 0.540764
\(85\) −111.299 −0.142024
\(86\) −35.8904 −0.0450019
\(87\) −106.206 −0.130879
\(88\) −2167.47 −2.62560
\(89\) −353.038 −0.420472 −0.210236 0.977651i \(-0.567423\pi\)
−0.210236 + 0.977651i \(0.567423\pi\)
\(90\) −500.846 −0.586597
\(91\) −261.093 −0.300769
\(92\) −2484.98 −2.81605
\(93\) 875.382 0.976053
\(94\) −1322.39 −1.45100
\(95\) −618.199 −0.667641
\(96\) −1199.91 −1.27568
\(97\) 1445.88 1.51347 0.756735 0.653722i \(-0.226793\pi\)
0.756735 + 0.653722i \(0.226793\pi\)
\(98\) −258.471 −0.266424
\(99\) 312.743 0.317493
\(100\) −271.619 −0.271619
\(101\) 474.852 0.467817 0.233909 0.972259i \(-0.424848\pi\)
0.233909 + 0.972259i \(0.424848\pi\)
\(102\) 166.949 0.162062
\(103\) −1999.59 −1.91287 −0.956433 0.291951i \(-0.905696\pi\)
−0.956433 + 0.291951i \(0.905696\pi\)
\(104\) 2326.51 2.19359
\(105\) 221.547 0.205912
\(106\) 2832.10 2.59508
\(107\) 1166.74 1.05414 0.527068 0.849823i \(-0.323291\pi\)
0.527068 + 0.849823i \(0.323291\pi\)
\(108\) 535.268 0.476909
\(109\) −1337.18 −1.17503 −0.587515 0.809213i \(-0.699894\pi\)
−0.587515 + 0.809213i \(0.699894\pi\)
\(110\) −1933.77 −1.67616
\(111\) −779.691 −0.666712
\(112\) 1192.96 1.00646
\(113\) 906.578 0.754723 0.377361 0.926066i \(-0.376831\pi\)
0.377361 + 0.926066i \(0.376831\pi\)
\(114\) 927.299 0.761838
\(115\) −1322.39 −1.07229
\(116\) −701.836 −0.561757
\(117\) −335.691 −0.265253
\(118\) 189.319 0.147697
\(119\) −73.8488 −0.0568883
\(120\) −1974.12 −1.50177
\(121\) −123.495 −0.0927836
\(122\) −304.859 −0.226235
\(123\) −1014.74 −0.743872
\(124\) 5784.74 4.18940
\(125\) −1463.27 −1.04703
\(126\) −332.320 −0.234963
\(127\) −1714.89 −1.19820 −0.599101 0.800674i \(-0.704475\pi\)
−0.599101 + 0.800674i \(0.704475\pi\)
\(128\) −737.564 −0.509313
\(129\) 20.4119 0.0139315
\(130\) 2075.67 1.40037
\(131\) 470.611 0.313874 0.156937 0.987609i \(-0.449838\pi\)
0.156937 + 0.987609i \(0.449838\pi\)
\(132\) 2066.68 1.36274
\(133\) −410.186 −0.267426
\(134\) −2540.88 −1.63805
\(135\) 284.846 0.181597
\(136\) 658.042 0.414901
\(137\) −443.910 −0.276831 −0.138415 0.990374i \(-0.544201\pi\)
−0.138415 + 0.990374i \(0.544201\pi\)
\(138\) 1983.59 1.22358
\(139\) 1669.98 1.01904 0.509518 0.860460i \(-0.329824\pi\)
0.509518 + 0.860460i \(0.329824\pi\)
\(140\) 1464.03 0.883811
\(141\) 752.083 0.449197
\(142\) −1918.76 −1.13394
\(143\) −1296.11 −0.757945
\(144\) 1533.80 0.887619
\(145\) −373.485 −0.213905
\(146\) −3066.30 −1.73814
\(147\) 147.000 0.0824786
\(148\) −5152.39 −2.86165
\(149\) 743.871 0.408995 0.204497 0.978867i \(-0.434444\pi\)
0.204497 + 0.978867i \(0.434444\pi\)
\(150\) 216.815 0.118019
\(151\) 606.764 0.327005 0.163503 0.986543i \(-0.447721\pi\)
0.163503 + 0.986543i \(0.447721\pi\)
\(152\) 3655.03 1.95041
\(153\) −94.9485 −0.0501708
\(154\) −1283.09 −0.671393
\(155\) 3078.38 1.59523
\(156\) −2218.33 −1.13852
\(157\) 3114.78 1.58336 0.791678 0.610939i \(-0.209208\pi\)
0.791678 + 0.610939i \(0.209208\pi\)
\(158\) 3659.16 1.84245
\(159\) −1610.70 −0.803377
\(160\) −4219.61 −2.08493
\(161\) −877.430 −0.429511
\(162\) −427.268 −0.207218
\(163\) 2413.07 1.15955 0.579774 0.814777i \(-0.303141\pi\)
0.579774 + 0.814777i \(0.303141\pi\)
\(164\) −6705.67 −3.19284
\(165\) 1099.79 0.518902
\(166\) −7038.81 −3.29107
\(167\) −610.475 −0.282874 −0.141437 0.989947i \(-0.545172\pi\)
−0.141437 + 0.989947i \(0.545172\pi\)
\(168\) −1309.87 −0.601538
\(169\) −805.784 −0.366766
\(170\) 587.093 0.264870
\(171\) −527.382 −0.235847
\(172\) 134.887 0.0597968
\(173\) 3793.81 1.66727 0.833636 0.552315i \(-0.186255\pi\)
0.833636 + 0.552315i \(0.186255\pi\)
\(174\) 560.228 0.244085
\(175\) −95.9070 −0.0414279
\(176\) 5922.05 2.53631
\(177\) −107.671 −0.0457235
\(178\) 1862.25 0.784165
\(179\) −2804.68 −1.17112 −0.585562 0.810627i \(-0.699126\pi\)
−0.585562 + 0.810627i \(0.699126\pi\)
\(180\) 1882.33 0.779448
\(181\) 3106.04 1.27553 0.637763 0.770232i \(-0.279860\pi\)
0.637763 + 0.770232i \(0.279860\pi\)
\(182\) 1377.24 0.560924
\(183\) 173.382 0.0700370
\(184\) 7818.48 3.13253
\(185\) −2741.87 −1.08966
\(186\) −4617.57 −1.82030
\(187\) −366.598 −0.143360
\(188\) 4969.95 1.92804
\(189\) 189.000 0.0727393
\(190\) 3260.95 1.24513
\(191\) 261.952 0.0992365 0.0496182 0.998768i \(-0.484200\pi\)
0.0496182 + 0.998768i \(0.484200\pi\)
\(192\) 2239.27 0.841694
\(193\) 4051.07 1.51089 0.755447 0.655210i \(-0.227420\pi\)
0.755447 + 0.655210i \(0.227420\pi\)
\(194\) −7626.88 −2.82257
\(195\) −1180.50 −0.433523
\(196\) 971.413 0.354013
\(197\) −2874.83 −1.03971 −0.519855 0.854254i \(-0.674014\pi\)
−0.519855 + 0.854254i \(0.674014\pi\)
\(198\) −1649.69 −0.592113
\(199\) −3066.97 −1.09252 −0.546261 0.837615i \(-0.683949\pi\)
−0.546261 + 0.837615i \(0.683949\pi\)
\(200\) 854.594 0.302145
\(201\) 1445.07 0.507103
\(202\) −2504.81 −0.872463
\(203\) −247.814 −0.0856804
\(204\) −627.444 −0.215342
\(205\) −3568.46 −1.21576
\(206\) 10547.7 3.56743
\(207\) −1128.12 −0.378793
\(208\) −6356.60 −2.11899
\(209\) −2036.23 −0.673919
\(210\) −1168.64 −0.384018
\(211\) 595.422 0.194268 0.0971340 0.995271i \(-0.469032\pi\)
0.0971340 + 0.995271i \(0.469032\pi\)
\(212\) −10643.9 −3.44824
\(213\) 1091.26 0.351041
\(214\) −6154.44 −1.96593
\(215\) 71.7808 0.0227693
\(216\) −1684.11 −0.530507
\(217\) 2042.56 0.638976
\(218\) 7053.49 2.19139
\(219\) 1743.90 0.538090
\(220\) 7267.71 2.22722
\(221\) 393.498 0.119772
\(222\) 4112.81 1.24339
\(223\) −3779.79 −1.13504 −0.567520 0.823360i \(-0.692097\pi\)
−0.567520 + 0.823360i \(0.692097\pi\)
\(224\) −2799.79 −0.835127
\(225\) −123.309 −0.0365360
\(226\) −4782.12 −1.40753
\(227\) 1827.62 0.534376 0.267188 0.963644i \(-0.413906\pi\)
0.267188 + 0.963644i \(0.413906\pi\)
\(228\) −3485.07 −1.01230
\(229\) −850.249 −0.245354 −0.122677 0.992447i \(-0.539148\pi\)
−0.122677 + 0.992447i \(0.539148\pi\)
\(230\) 6975.51 1.99979
\(231\) 729.733 0.207848
\(232\) 2208.18 0.624890
\(233\) −6591.10 −1.85321 −0.926604 0.376039i \(-0.877286\pi\)
−0.926604 + 0.376039i \(0.877286\pi\)
\(234\) 1770.74 0.494688
\(235\) 2644.78 0.734156
\(236\) −711.518 −0.196254
\(237\) −2081.07 −0.570381
\(238\) 389.547 0.106095
\(239\) −182.556 −0.0494083 −0.0247042 0.999695i \(-0.507864\pi\)
−0.0247042 + 0.999695i \(0.507864\pi\)
\(240\) 5393.80 1.45070
\(241\) 1523.90 0.407315 0.203657 0.979042i \(-0.434717\pi\)
0.203657 + 0.979042i \(0.434717\pi\)
\(242\) 651.426 0.173038
\(243\) 243.000 0.0641500
\(244\) 1145.75 0.300612
\(245\) 516.942 0.134801
\(246\) 5352.68 1.38730
\(247\) 2185.65 0.563034
\(248\) −18200.5 −4.66022
\(249\) 4003.18 1.01884
\(250\) 7718.64 1.95268
\(251\) 2357.73 0.592903 0.296451 0.955048i \(-0.404197\pi\)
0.296451 + 0.955048i \(0.404197\pi\)
\(252\) 1248.96 0.312210
\(253\) −4355.71 −1.08238
\(254\) 9045.89 2.23460
\(255\) −333.897 −0.0819978
\(256\) −2080.79 −0.508006
\(257\) 2782.55 0.675372 0.337686 0.941259i \(-0.390356\pi\)
0.337686 + 0.941259i \(0.390356\pi\)
\(258\) −107.671 −0.0259818
\(259\) −1819.28 −0.436465
\(260\) −7801.01 −1.86076
\(261\) −318.618 −0.0755630
\(262\) −2482.44 −0.585364
\(263\) 2043.78 0.479183 0.239591 0.970874i \(-0.422987\pi\)
0.239591 + 0.970874i \(0.422987\pi\)
\(264\) −6502.40 −1.51589
\(265\) −5664.21 −1.31302
\(266\) 2163.70 0.498740
\(267\) −1059.11 −0.242759
\(268\) 9549.41 2.17658
\(269\) 3452.84 0.782614 0.391307 0.920260i \(-0.372023\pi\)
0.391307 + 0.920260i \(0.372023\pi\)
\(270\) −1502.54 −0.338672
\(271\) 2644.29 0.592728 0.296364 0.955075i \(-0.404226\pi\)
0.296364 + 0.955075i \(0.404226\pi\)
\(272\) −1797.93 −0.400793
\(273\) −783.279 −0.173649
\(274\) 2341.59 0.516280
\(275\) −476.098 −0.104399
\(276\) −7454.93 −1.62585
\(277\) 2679.49 0.581208 0.290604 0.956843i \(-0.406144\pi\)
0.290604 + 0.956843i \(0.406144\pi\)
\(278\) −8809.01 −1.90046
\(279\) 2626.15 0.563524
\(280\) −4606.29 −0.983138
\(281\) −1019.69 −0.216476 −0.108238 0.994125i \(-0.534521\pi\)
−0.108238 + 0.994125i \(0.534521\pi\)
\(282\) −3967.18 −0.837737
\(283\) 432.206 0.0907844 0.0453922 0.998969i \(-0.485546\pi\)
0.0453922 + 0.998969i \(0.485546\pi\)
\(284\) 7211.30 1.50673
\(285\) −1854.60 −0.385463
\(286\) 6836.87 1.41354
\(287\) −2367.73 −0.486979
\(288\) −3599.72 −0.736513
\(289\) −4801.70 −0.977346
\(290\) 1970.10 0.398926
\(291\) 4337.63 0.873802
\(292\) 11524.1 2.30958
\(293\) −2245.92 −0.447809 −0.223904 0.974611i \(-0.571880\pi\)
−0.223904 + 0.974611i \(0.571880\pi\)
\(294\) −775.413 −0.153820
\(295\) −378.638 −0.0747293
\(296\) 16211.0 3.18325
\(297\) 938.228 0.183305
\(298\) −3923.86 −0.762761
\(299\) 4675.33 0.904284
\(300\) −814.856 −0.156819
\(301\) 47.6278 0.00912034
\(302\) −3200.63 −0.609853
\(303\) 1424.56 0.270094
\(304\) −9986.44 −1.88408
\(305\) 609.718 0.114467
\(306\) 500.846 0.0935668
\(307\) −3197.08 −0.594354 −0.297177 0.954822i \(-0.596045\pi\)
−0.297177 + 0.954822i \(0.596045\pi\)
\(308\) 4822.26 0.892122
\(309\) −5998.76 −1.10439
\(310\) −16238.2 −2.97506
\(311\) −3355.60 −0.611829 −0.305915 0.952059i \(-0.598962\pi\)
−0.305915 + 0.952059i \(0.598962\pi\)
\(312\) 6979.53 1.26647
\(313\) −2256.39 −0.407472 −0.203736 0.979026i \(-0.565308\pi\)
−0.203736 + 0.979026i \(0.565308\pi\)
\(314\) −16430.2 −2.95290
\(315\) 664.640 0.118883
\(316\) −13752.3 −2.44818
\(317\) −6139.19 −1.08773 −0.543866 0.839172i \(-0.683040\pi\)
−0.543866 + 0.839172i \(0.683040\pi\)
\(318\) 8496.31 1.49827
\(319\) −1230.19 −0.215917
\(320\) 7874.64 1.37564
\(321\) 3500.21 0.608606
\(322\) 4628.37 0.801022
\(323\) 618.199 0.106494
\(324\) 1605.80 0.275344
\(325\) 511.033 0.0872216
\(326\) −12728.8 −2.16252
\(327\) −4011.53 −0.678404
\(328\) 21098.0 3.55166
\(329\) 1754.86 0.294069
\(330\) −5801.32 −0.967734
\(331\) 7029.81 1.16735 0.583676 0.811987i \(-0.301614\pi\)
0.583676 + 0.811987i \(0.301614\pi\)
\(332\) 26454.0 4.37305
\(333\) −2339.07 −0.384926
\(334\) 3220.21 0.527550
\(335\) 5081.76 0.828795
\(336\) 3578.88 0.581083
\(337\) 10328.4 1.66951 0.834757 0.550619i \(-0.185608\pi\)
0.834757 + 0.550619i \(0.185608\pi\)
\(338\) 4250.44 0.684005
\(339\) 2719.73 0.435740
\(340\) −2206.48 −0.351950
\(341\) 10139.6 1.61024
\(342\) 2781.90 0.439847
\(343\) 343.000 0.0539949
\(344\) −424.395 −0.0665170
\(345\) −3967.18 −0.619089
\(346\) −20012.0 −3.10940
\(347\) 1967.54 0.304389 0.152194 0.988351i \(-0.451366\pi\)
0.152194 + 0.988351i \(0.451366\pi\)
\(348\) −2105.51 −0.324330
\(349\) −4365.46 −0.669564 −0.334782 0.942296i \(-0.608663\pi\)
−0.334782 + 0.942296i \(0.608663\pi\)
\(350\) 505.901 0.0772616
\(351\) −1007.07 −0.153144
\(352\) −13898.6 −2.10454
\(353\) −6071.59 −0.915462 −0.457731 0.889091i \(-0.651338\pi\)
−0.457731 + 0.889091i \(0.651338\pi\)
\(354\) 567.957 0.0852728
\(355\) 3837.53 0.573732
\(356\) −6998.90 −1.04197
\(357\) −221.547 −0.0328445
\(358\) 14794.4 2.18411
\(359\) 9638.04 1.41693 0.708463 0.705748i \(-0.249389\pi\)
0.708463 + 0.705748i \(0.249389\pi\)
\(360\) −5922.37 −0.867046
\(361\) −3425.27 −0.499384
\(362\) −16384.1 −2.37881
\(363\) −370.485 −0.0535687
\(364\) −5176.10 −0.745334
\(365\) 6132.61 0.879439
\(366\) −914.576 −0.130617
\(367\) 522.725 0.0743488 0.0371744 0.999309i \(-0.488164\pi\)
0.0371744 + 0.999309i \(0.488164\pi\)
\(368\) −21362.0 −3.02601
\(369\) −3044.23 −0.429475
\(370\) 14463.1 2.03217
\(371\) −3758.30 −0.525934
\(372\) 17354.2 2.41875
\(373\) 3229.84 0.448351 0.224175 0.974549i \(-0.428031\pi\)
0.224175 + 0.974549i \(0.428031\pi\)
\(374\) 1933.77 0.267361
\(375\) −4389.82 −0.604505
\(376\) −15637.0 −2.14472
\(377\) 1320.46 0.180390
\(378\) −996.959 −0.135656
\(379\) 6639.71 0.899892 0.449946 0.893056i \(-0.351443\pi\)
0.449946 + 0.893056i \(0.351443\pi\)
\(380\) −12255.6 −1.65448
\(381\) −5144.66 −0.691782
\(382\) −1381.77 −0.185073
\(383\) −14224.4 −1.89774 −0.948871 0.315664i \(-0.897773\pi\)
−0.948871 + 0.315664i \(0.897773\pi\)
\(384\) −2212.69 −0.294052
\(385\) 2566.19 0.339701
\(386\) −21369.1 −2.81777
\(387\) 61.2358 0.00804338
\(388\) 28664.2 3.75052
\(389\) 2921.82 0.380828 0.190414 0.981704i \(-0.439017\pi\)
0.190414 + 0.981704i \(0.439017\pi\)
\(390\) 6227.01 0.808505
\(391\) 1322.39 0.171039
\(392\) −3056.35 −0.393799
\(393\) 1411.83 0.181215
\(394\) 15164.5 1.93902
\(395\) −7318.33 −0.932215
\(396\) 6200.04 0.786778
\(397\) 811.940 0.102645 0.0513226 0.998682i \(-0.483656\pi\)
0.0513226 + 0.998682i \(0.483656\pi\)
\(398\) 16178.0 2.03751
\(399\) −1230.56 −0.154398
\(400\) −2334.96 −0.291870
\(401\) 2338.63 0.291237 0.145618 0.989341i \(-0.453483\pi\)
0.145618 + 0.989341i \(0.453483\pi\)
\(402\) −7622.64 −0.945728
\(403\) −10883.6 −1.34529
\(404\) 9413.83 1.15930
\(405\) 854.537 0.104845
\(406\) 1307.20 0.159791
\(407\) −9031.21 −1.09990
\(408\) 1974.12 0.239543
\(409\) −2727.57 −0.329755 −0.164877 0.986314i \(-0.552723\pi\)
−0.164877 + 0.986314i \(0.552723\pi\)
\(410\) 18823.3 2.26736
\(411\) −1331.73 −0.159828
\(412\) −39641.3 −4.74026
\(413\) −251.233 −0.0299331
\(414\) 5950.76 0.706435
\(415\) 14077.6 1.66516
\(416\) 14918.5 1.75826
\(417\) 5009.94 0.588340
\(418\) 10741.0 1.25684
\(419\) 13306.3 1.55144 0.775721 0.631076i \(-0.217386\pi\)
0.775721 + 0.631076i \(0.217386\pi\)
\(420\) 4392.10 0.510268
\(421\) −11007.5 −1.27428 −0.637138 0.770750i \(-0.719882\pi\)
−0.637138 + 0.770750i \(0.719882\pi\)
\(422\) −3140.80 −0.362303
\(423\) 2256.25 0.259344
\(424\) 33488.9 3.83577
\(425\) 144.543 0.0164974
\(426\) −5756.29 −0.654679
\(427\) 404.558 0.0458500
\(428\) 23130.2 2.61225
\(429\) −3888.33 −0.437600
\(430\) −378.638 −0.0424640
\(431\) −6525.62 −0.729300 −0.364650 0.931145i \(-0.618811\pi\)
−0.364650 + 0.931145i \(0.618811\pi\)
\(432\) 4601.41 0.512467
\(433\) −11716.3 −1.30034 −0.650171 0.759788i \(-0.725303\pi\)
−0.650171 + 0.759788i \(0.725303\pi\)
\(434\) −10774.3 −1.19167
\(435\) −1120.46 −0.123498
\(436\) −26509.2 −2.91183
\(437\) 7345.10 0.804036
\(438\) −9198.91 −1.00352
\(439\) −14611.4 −1.58853 −0.794264 0.607573i \(-0.792143\pi\)
−0.794264 + 0.607573i \(0.792143\pi\)
\(440\) −22866.4 −2.47753
\(441\) 441.000 0.0476190
\(442\) −2075.67 −0.223370
\(443\) −15239.8 −1.63446 −0.817228 0.576314i \(-0.804490\pi\)
−0.817228 + 0.576314i \(0.804490\pi\)
\(444\) −15457.2 −1.65217
\(445\) −3724.50 −0.396760
\(446\) 19938.1 2.11681
\(447\) 2231.61 0.236133
\(448\) 5224.96 0.551018
\(449\) 10678.8 1.12241 0.561206 0.827676i \(-0.310338\pi\)
0.561206 + 0.827676i \(0.310338\pi\)
\(450\) 650.444 0.0681383
\(451\) −11753.8 −1.22720
\(452\) 17972.7 1.87027
\(453\) 1820.29 0.188796
\(454\) −9640.53 −0.996592
\(455\) −2754.49 −0.283807
\(456\) 10965.1 1.12607
\(457\) 4228.23 0.432797 0.216399 0.976305i \(-0.430569\pi\)
0.216399 + 0.976305i \(0.430569\pi\)
\(458\) 4484.99 0.457577
\(459\) −284.846 −0.0289661
\(460\) −26216.1 −2.65724
\(461\) 910.121 0.0919492 0.0459746 0.998943i \(-0.485361\pi\)
0.0459746 + 0.998943i \(0.485361\pi\)
\(462\) −3849.28 −0.387629
\(463\) 4456.16 0.447290 0.223645 0.974671i \(-0.428204\pi\)
0.223645 + 0.974671i \(0.428204\pi\)
\(464\) −6033.30 −0.603640
\(465\) 9235.14 0.921009
\(466\) 34767.5 3.45617
\(467\) −4429.42 −0.438907 −0.219453 0.975623i \(-0.570427\pi\)
−0.219453 + 0.975623i \(0.570427\pi\)
\(468\) −6654.99 −0.657323
\(469\) 3371.84 0.331977
\(470\) −13951.0 −1.36918
\(471\) 9344.35 0.914151
\(472\) 2238.65 0.218310
\(473\) 236.432 0.0229835
\(474\) 10977.5 1.06374
\(475\) 802.851 0.0775523
\(476\) −1464.03 −0.140975
\(477\) −4832.10 −0.463830
\(478\) 962.970 0.0921448
\(479\) 2752.85 0.262591 0.131296 0.991343i \(-0.458086\pi\)
0.131296 + 0.991343i \(0.458086\pi\)
\(480\) −12658.8 −1.20374
\(481\) 9693.90 0.918927
\(482\) −8038.43 −0.759628
\(483\) −2632.29 −0.247978
\(484\) −2448.26 −0.229927
\(485\) 15253.8 1.42812
\(486\) −1281.80 −0.119638
\(487\) −670.598 −0.0623977 −0.0311989 0.999513i \(-0.509933\pi\)
−0.0311989 + 0.999513i \(0.509933\pi\)
\(488\) −3604.88 −0.334396
\(489\) 7239.22 0.669466
\(490\) −2726.83 −0.251399
\(491\) −8244.70 −0.757797 −0.378898 0.925438i \(-0.623697\pi\)
−0.378898 + 0.925438i \(0.623697\pi\)
\(492\) −20117.0 −1.84338
\(493\) 373.485 0.0341195
\(494\) −11529.1 −1.05004
\(495\) 3299.38 0.299588
\(496\) 49728.3 4.50175
\(497\) 2546.27 0.229810
\(498\) −21116.4 −1.90010
\(499\) 8164.91 0.732488 0.366244 0.930519i \(-0.380644\pi\)
0.366244 + 0.930519i \(0.380644\pi\)
\(500\) −29009.0 −2.59465
\(501\) −1831.43 −0.163317
\(502\) −12436.8 −1.10574
\(503\) 8175.59 0.724715 0.362357 0.932039i \(-0.381972\pi\)
0.362357 + 0.932039i \(0.381972\pi\)
\(504\) −3929.60 −0.347298
\(505\) 5009.61 0.441435
\(506\) 22976.0 2.01859
\(507\) −2417.35 −0.211752
\(508\) −33997.2 −2.96926
\(509\) −878.448 −0.0764961 −0.0382480 0.999268i \(-0.512178\pi\)
−0.0382480 + 0.999268i \(0.512178\pi\)
\(510\) 1761.28 0.152923
\(511\) 4069.09 0.352262
\(512\) 16876.5 1.45673
\(513\) −1582.15 −0.136167
\(514\) −14677.7 −1.25955
\(515\) −21095.3 −1.80499
\(516\) 404.661 0.0345237
\(517\) 8711.42 0.741060
\(518\) 9596.55 0.813992
\(519\) 11381.4 0.962600
\(520\) 24544.3 2.06988
\(521\) 11712.6 0.984910 0.492455 0.870338i \(-0.336100\pi\)
0.492455 + 0.870338i \(0.336100\pi\)
\(522\) 1680.68 0.140922
\(523\) −7341.82 −0.613834 −0.306917 0.951736i \(-0.599297\pi\)
−0.306917 + 0.951736i \(0.599297\pi\)
\(524\) 9329.75 0.777809
\(525\) −287.721 −0.0239184
\(526\) −10780.8 −0.893659
\(527\) −3078.38 −0.254452
\(528\) 17766.1 1.46434
\(529\) 3544.92 0.291355
\(530\) 29878.2 2.44873
\(531\) −323.014 −0.0263985
\(532\) −8131.84 −0.662707
\(533\) 12616.3 1.02528
\(534\) 5586.74 0.452738
\(535\) 12308.9 0.994690
\(536\) −30045.3 −2.42119
\(537\) −8414.03 −0.676149
\(538\) −18213.4 −1.45955
\(539\) 1702.71 0.136068
\(540\) 5646.99 0.450015
\(541\) −15868.7 −1.26109 −0.630545 0.776153i \(-0.717169\pi\)
−0.630545 + 0.776153i \(0.717169\pi\)
\(542\) −13948.4 −1.10542
\(543\) 9318.13 0.736426
\(544\) 4219.61 0.332563
\(545\) −14107.0 −1.10877
\(546\) 4131.73 0.323850
\(547\) 2315.26 0.180975 0.0904875 0.995898i \(-0.471157\pi\)
0.0904875 + 0.995898i \(0.471157\pi\)
\(548\) −8800.41 −0.686013
\(549\) 520.146 0.0404359
\(550\) 2511.38 0.194701
\(551\) 2074.49 0.160392
\(552\) 23455.4 1.80857
\(553\) −4855.84 −0.373402
\(554\) −14134.1 −1.08393
\(555\) −8225.61 −0.629113
\(556\) 33106.9 2.52526
\(557\) −4819.05 −0.366588 −0.183294 0.983058i \(-0.558676\pi\)
−0.183294 + 0.983058i \(0.558676\pi\)
\(558\) −13852.7 −1.05095
\(559\) −253.781 −0.0192018
\(560\) 12585.5 0.949706
\(561\) −1099.79 −0.0827689
\(562\) 5378.79 0.403720
\(563\) 2540.86 0.190203 0.0951017 0.995468i \(-0.469682\pi\)
0.0951017 + 0.995468i \(0.469682\pi\)
\(564\) 14909.9 1.11315
\(565\) 9564.25 0.712161
\(566\) −2279.85 −0.169310
\(567\) 567.000 0.0419961
\(568\) −22688.9 −1.67607
\(569\) −24220.0 −1.78445 −0.892227 0.451587i \(-0.850858\pi\)
−0.892227 + 0.451587i \(0.850858\pi\)
\(570\) 9782.85 0.718875
\(571\) −11772.1 −0.862778 −0.431389 0.902166i \(-0.641976\pi\)
−0.431389 + 0.902166i \(0.641976\pi\)
\(572\) −25695.1 −1.87826
\(573\) 785.855 0.0572942
\(574\) 12489.6 0.908198
\(575\) 1717.38 0.124556
\(576\) 6717.80 0.485952
\(577\) 10584.3 0.763655 0.381827 0.924234i \(-0.375295\pi\)
0.381827 + 0.924234i \(0.375295\pi\)
\(578\) 25328.6 1.82272
\(579\) 12153.2 0.872315
\(580\) −7404.25 −0.530077
\(581\) 9340.74 0.666987
\(582\) −22880.6 −1.62961
\(583\) −18656.8 −1.32536
\(584\) −36258.3 −2.56914
\(585\) −3541.49 −0.250295
\(586\) 11847.0 0.835148
\(587\) −8712.63 −0.612621 −0.306311 0.951932i \(-0.599095\pi\)
−0.306311 + 0.951932i \(0.599095\pi\)
\(588\) 2914.24 0.204390
\(589\) −17098.6 −1.19615
\(590\) 1997.28 0.139368
\(591\) −8624.48 −0.600277
\(592\) −44292.4 −3.07501
\(593\) −15362.9 −1.06387 −0.531937 0.846784i \(-0.678536\pi\)
−0.531937 + 0.846784i \(0.678536\pi\)
\(594\) −4949.07 −0.341857
\(595\) −779.093 −0.0536802
\(596\) 14747.0 1.01353
\(597\) −9200.91 −0.630768
\(598\) −24662.0 −1.68646
\(599\) 26003.8 1.77377 0.886883 0.461994i \(-0.152866\pi\)
0.886883 + 0.461994i \(0.152866\pi\)
\(600\) 2563.78 0.174443
\(601\) 20567.7 1.39596 0.697982 0.716115i \(-0.254082\pi\)
0.697982 + 0.716115i \(0.254082\pi\)
\(602\) −251.233 −0.0170091
\(603\) 4335.22 0.292776
\(604\) 12029.0 0.810349
\(605\) −1302.85 −0.0875512
\(606\) −7514.42 −0.503717
\(607\) 19642.1 1.31342 0.656711 0.754142i \(-0.271947\pi\)
0.656711 + 0.754142i \(0.271947\pi\)
\(608\) 23437.4 1.56334
\(609\) −743.442 −0.0494676
\(610\) −3216.21 −0.213476
\(611\) −9350.65 −0.619127
\(612\) −1882.33 −0.124328
\(613\) 8454.59 0.557060 0.278530 0.960428i \(-0.410153\pi\)
0.278530 + 0.960428i \(0.410153\pi\)
\(614\) 16864.3 1.10845
\(615\) −10705.4 −0.701922
\(616\) −15172.3 −0.992383
\(617\) −24168.4 −1.57696 −0.788479 0.615061i \(-0.789131\pi\)
−0.788479 + 0.615061i \(0.789131\pi\)
\(618\) 31643.0 2.05966
\(619\) −2037.56 −0.132305 −0.0661523 0.997810i \(-0.521072\pi\)
−0.0661523 + 0.997810i \(0.521072\pi\)
\(620\) 61028.1 3.95314
\(621\) −3384.37 −0.218696
\(622\) 17700.5 1.14104
\(623\) −2471.27 −0.158923
\(624\) −19069.8 −1.22340
\(625\) −13724.7 −0.878378
\(626\) 11902.3 0.759921
\(627\) −6108.70 −0.389088
\(628\) 61749.8 3.92370
\(629\) 2741.87 0.173808
\(630\) −3505.92 −0.221713
\(631\) 12339.5 0.778489 0.389244 0.921135i \(-0.372736\pi\)
0.389244 + 0.921135i \(0.372736\pi\)
\(632\) 43268.7 2.72332
\(633\) 1786.27 0.112161
\(634\) 32383.7 2.02858
\(635\) −18091.8 −1.13063
\(636\) −31931.7 −1.99084
\(637\) −1827.65 −0.113680
\(638\) 6489.15 0.402677
\(639\) 3273.77 0.202674
\(640\) −7781.18 −0.480591
\(641\) −10222.6 −0.629906 −0.314953 0.949107i \(-0.601989\pi\)
−0.314953 + 0.949107i \(0.601989\pi\)
\(642\) −18463.3 −1.13503
\(643\) −1211.75 −0.0743187 −0.0371594 0.999309i \(-0.511831\pi\)
−0.0371594 + 0.999309i \(0.511831\pi\)
\(644\) −17394.8 −1.06437
\(645\) 215.342 0.0131459
\(646\) −3260.95 −0.198607
\(647\) −2817.22 −0.171184 −0.0855922 0.996330i \(-0.527278\pi\)
−0.0855922 + 0.996330i \(0.527278\pi\)
\(648\) −5052.34 −0.306288
\(649\) −1247.16 −0.0754320
\(650\) −2695.66 −0.162665
\(651\) 6127.67 0.368913
\(652\) 47838.6 2.87347
\(653\) 20986.2 1.25766 0.628831 0.777542i \(-0.283534\pi\)
0.628831 + 0.777542i \(0.283534\pi\)
\(654\) 21160.5 1.26520
\(655\) 4964.87 0.296173
\(656\) −57645.1 −3.43089
\(657\) 5231.69 0.310666
\(658\) −9256.74 −0.548428
\(659\) −2384.09 −0.140927 −0.0704635 0.997514i \(-0.522448\pi\)
−0.0704635 + 0.997514i \(0.522448\pi\)
\(660\) 21803.1 1.28589
\(661\) −7577.10 −0.445862 −0.222931 0.974834i \(-0.571562\pi\)
−0.222931 + 0.974834i \(0.571562\pi\)
\(662\) −37081.7 −2.17707
\(663\) 1180.50 0.0691503
\(664\) −83232.2 −4.86451
\(665\) −4327.40 −0.252345
\(666\) 12338.4 0.717874
\(667\) 4437.54 0.257605
\(668\) −12102.5 −0.700989
\(669\) −11339.4 −0.655315
\(670\) −26805.9 −1.54567
\(671\) 2008.30 0.115543
\(672\) −8399.36 −0.482161
\(673\) 11724.6 0.671547 0.335774 0.941943i \(-0.391002\pi\)
0.335774 + 0.941943i \(0.391002\pi\)
\(674\) −54481.7 −3.11358
\(675\) −369.927 −0.0210941
\(676\) −15974.5 −0.908880
\(677\) 32304.3 1.83390 0.916952 0.398997i \(-0.130642\pi\)
0.916952 + 0.398997i \(0.130642\pi\)
\(678\) −14346.4 −0.812639
\(679\) 10121.1 0.572038
\(680\) 6942.23 0.391503
\(681\) 5482.85 0.308522
\(682\) −53485.6 −3.00303
\(683\) 33367.1 1.86934 0.934669 0.355519i \(-0.115696\pi\)
0.934669 + 0.355519i \(0.115696\pi\)
\(684\) −10455.2 −0.584452
\(685\) −4683.18 −0.261219
\(686\) −1809.30 −0.100699
\(687\) −2550.75 −0.141655
\(688\) 1159.55 0.0642551
\(689\) 20025.8 1.10729
\(690\) 20926.5 1.15458
\(691\) −1043.67 −0.0574577 −0.0287288 0.999587i \(-0.509146\pi\)
−0.0287288 + 0.999587i \(0.509146\pi\)
\(692\) 75211.3 4.13166
\(693\) 2189.20 0.120001
\(694\) −10378.6 −0.567674
\(695\) 17618.0 0.961567
\(696\) 6624.55 0.360780
\(697\) 3568.46 0.193924
\(698\) 23027.4 1.24871
\(699\) −19773.3 −1.06995
\(700\) −1901.33 −0.102662
\(701\) −11305.7 −0.609143 −0.304572 0.952489i \(-0.598513\pi\)
−0.304572 + 0.952489i \(0.598513\pi\)
\(702\) 5312.23 0.285608
\(703\) 15229.4 0.817055
\(704\) 25937.6 1.38858
\(705\) 7934.35 0.423865
\(706\) 32027.1 1.70730
\(707\) 3323.97 0.176818
\(708\) −2134.55 −0.113307
\(709\) −13306.8 −0.704860 −0.352430 0.935838i \(-0.614645\pi\)
−0.352430 + 0.935838i \(0.614645\pi\)
\(710\) −20242.6 −1.06999
\(711\) −6243.22 −0.329309
\(712\) 22020.6 1.15907
\(713\) −36575.6 −1.92113
\(714\) 1168.64 0.0612538
\(715\) −13673.7 −0.715201
\(716\) −55602.0 −2.90216
\(717\) −547.669 −0.0285259
\(718\) −50839.9 −2.64252
\(719\) 10701.2 0.555062 0.277531 0.960717i \(-0.410484\pi\)
0.277531 + 0.960717i \(0.410484\pi\)
\(720\) 16181.4 0.837562
\(721\) −13997.1 −0.722996
\(722\) 18068.0 0.931333
\(723\) 4571.69 0.235163
\(724\) 61576.5 3.16088
\(725\) 485.042 0.0248469
\(726\) 1954.28 0.0999037
\(727\) −2121.14 −0.108210 −0.0541051 0.998535i \(-0.517231\pi\)
−0.0541051 + 0.998535i \(0.517231\pi\)
\(728\) 16285.6 0.829098
\(729\) 729.000 0.0370370
\(730\) −32349.0 −1.64012
\(731\) −71.7808 −0.00363189
\(732\) 3437.26 0.173558
\(733\) −21584.0 −1.08762 −0.543809 0.839209i \(-0.683019\pi\)
−0.543809 + 0.839209i \(0.683019\pi\)
\(734\) −2757.33 −0.138658
\(735\) 1550.83 0.0778273
\(736\) 50135.0 2.51087
\(737\) 16738.4 0.836588
\(738\) 16058.0 0.800955
\(739\) −9945.21 −0.495048 −0.247524 0.968882i \(-0.579617\pi\)
−0.247524 + 0.968882i \(0.579617\pi\)
\(740\) −54356.9 −2.70027
\(741\) 6556.94 0.325068
\(742\) 19824.7 0.980848
\(743\) 2867.01 0.141562 0.0707808 0.997492i \(-0.477451\pi\)
0.0707808 + 0.997492i \(0.477451\pi\)
\(744\) −54601.6 −2.69058
\(745\) 7847.71 0.385930
\(746\) −17037.1 −0.836158
\(747\) 12009.5 0.588227
\(748\) −7267.71 −0.355259
\(749\) 8167.15 0.398426
\(750\) 23155.9 1.12738
\(751\) −10824.1 −0.525934 −0.262967 0.964805i \(-0.584701\pi\)
−0.262967 + 0.964805i \(0.584701\pi\)
\(752\) 42724.0 2.07179
\(753\) 7073.19 0.342313
\(754\) −6965.31 −0.336421
\(755\) 6401.26 0.308564
\(756\) 3746.88 0.180255
\(757\) −14512.0 −0.696761 −0.348381 0.937353i \(-0.613268\pi\)
−0.348381 + 0.937353i \(0.613268\pi\)
\(758\) −35023.9 −1.67827
\(759\) −13067.1 −0.624910
\(760\) 38559.9 1.84042
\(761\) −33075.8 −1.57556 −0.787778 0.615959i \(-0.788769\pi\)
−0.787778 + 0.615959i \(0.788769\pi\)
\(762\) 27137.7 1.29015
\(763\) −9360.23 −0.444120
\(764\) 5193.13 0.245917
\(765\) −1001.69 −0.0473415
\(766\) 75032.8 3.53922
\(767\) 1338.68 0.0630206
\(768\) −6242.38 −0.293297
\(769\) 6728.44 0.315518 0.157759 0.987478i \(-0.449573\pi\)
0.157759 + 0.987478i \(0.449573\pi\)
\(770\) −13536.4 −0.633531
\(771\) 8347.65 0.389926
\(772\) 80311.5 3.74414
\(773\) −24233.3 −1.12757 −0.563784 0.825922i \(-0.690655\pi\)
−0.563784 + 0.825922i \(0.690655\pi\)
\(774\) −323.014 −0.0150006
\(775\) −3997.87 −0.185300
\(776\) −90186.0 −4.17202
\(777\) −5457.84 −0.251993
\(778\) −15412.4 −0.710231
\(779\) 19820.6 0.911615
\(780\) −23403.0 −1.07431
\(781\) 12640.1 0.579127
\(782\) −6975.51 −0.318982
\(783\) −955.854 −0.0436263
\(784\) 8350.72 0.380408
\(785\) 32860.5 1.49406
\(786\) −7447.31 −0.337960
\(787\) −17200.4 −0.779069 −0.389535 0.921012i \(-0.627364\pi\)
−0.389535 + 0.921012i \(0.627364\pi\)
\(788\) −56992.7 −2.57650
\(789\) 6131.35 0.276656
\(790\) 38603.6 1.73855
\(791\) 6346.05 0.285258
\(792\) −19507.2 −0.875199
\(793\) −2155.66 −0.0965318
\(794\) −4282.92 −0.191430
\(795\) −16992.6 −0.758071
\(796\) −60801.9 −2.70737
\(797\) −4208.87 −0.187059 −0.0935295 0.995617i \(-0.529815\pi\)
−0.0935295 + 0.995617i \(0.529815\pi\)
\(798\) 6491.09 0.287948
\(799\) −2644.78 −0.117104
\(800\) 5479.98 0.242183
\(801\) −3177.34 −0.140157
\(802\) −12336.1 −0.543146
\(803\) 20199.7 0.887709
\(804\) 28648.2 1.25665
\(805\) −9256.74 −0.405289
\(806\) 57410.2 2.50892
\(807\) 10358.5 0.451842
\(808\) −29618.7 −1.28958
\(809\) −23632.1 −1.02702 −0.513511 0.858083i \(-0.671656\pi\)
−0.513511 + 0.858083i \(0.671656\pi\)
\(810\) −4507.61 −0.195532
\(811\) 28425.1 1.23075 0.615377 0.788233i \(-0.289004\pi\)
0.615377 + 0.788233i \(0.289004\pi\)
\(812\) −4912.85 −0.212324
\(813\) 7932.88 0.342212
\(814\) 47638.9 2.05128
\(815\) 25457.5 1.09416
\(816\) −5393.80 −0.231398
\(817\) −398.699 −0.0170731
\(818\) 14387.7 0.614981
\(819\) −2349.84 −0.100256
\(820\) −70743.7 −3.01278
\(821\) 39409.6 1.67528 0.837640 0.546223i \(-0.183935\pi\)
0.837640 + 0.546223i \(0.183935\pi\)
\(822\) 7024.77 0.298074
\(823\) 16346.6 0.692352 0.346176 0.938170i \(-0.387480\pi\)
0.346176 + 0.938170i \(0.387480\pi\)
\(824\) 124723. 5.27300
\(825\) −1428.29 −0.0602749
\(826\) 1325.23 0.0558241
\(827\) −3738.87 −0.157211 −0.0786054 0.996906i \(-0.525047\pi\)
−0.0786054 + 0.996906i \(0.525047\pi\)
\(828\) −22364.8 −0.938684
\(829\) −45196.2 −1.89352 −0.946761 0.321937i \(-0.895666\pi\)
−0.946761 + 0.321937i \(0.895666\pi\)
\(830\) −74258.3 −3.10547
\(831\) 8038.46 0.335561
\(832\) −27840.8 −1.16010
\(833\) −516.942 −0.0215018
\(834\) −26427.0 −1.09723
\(835\) −6440.41 −0.266922
\(836\) −40367.8 −1.67004
\(837\) 7878.44 0.325351
\(838\) −70189.5 −2.89338
\(839\) 15899.7 0.654254 0.327127 0.944980i \(-0.393920\pi\)
0.327127 + 0.944980i \(0.393920\pi\)
\(840\) −13818.9 −0.567615
\(841\) −23135.7 −0.948612
\(842\) 58063.4 2.37648
\(843\) −3059.07 −0.124982
\(844\) 11804.1 0.481414
\(845\) −8500.89 −0.346082
\(846\) −11901.5 −0.483668
\(847\) −864.465 −0.0350689
\(848\) −91500.0 −3.70534
\(849\) 1296.62 0.0524144
\(850\) −762.453 −0.0307670
\(851\) 32577.4 1.31227
\(852\) 21633.9 0.869913
\(853\) 33926.7 1.36182 0.680908 0.732369i \(-0.261585\pi\)
0.680908 + 0.732369i \(0.261585\pi\)
\(854\) −2134.01 −0.0855086
\(855\) −5563.79 −0.222547
\(856\) −72774.7 −2.90583
\(857\) −35432.4 −1.41231 −0.706154 0.708058i \(-0.749572\pi\)
−0.706154 + 0.708058i \(0.749572\pi\)
\(858\) 20510.6 0.816108
\(859\) −6780.17 −0.269309 −0.134655 0.990893i \(-0.542992\pi\)
−0.134655 + 0.990893i \(0.542992\pi\)
\(860\) 1423.04 0.0564246
\(861\) −7103.20 −0.281157
\(862\) 34422.1 1.36012
\(863\) −30675.1 −1.20995 −0.604977 0.796243i \(-0.706818\pi\)
−0.604977 + 0.796243i \(0.706818\pi\)
\(864\) −10799.2 −0.425226
\(865\) 40024.1 1.57325
\(866\) 61802.4 2.42509
\(867\) −14405.1 −0.564271
\(868\) 40493.2 1.58344
\(869\) −24105.2 −0.940981
\(870\) 5910.31 0.230320
\(871\) −17966.6 −0.698938
\(872\) 83405.8 3.23908
\(873\) 13012.9 0.504490
\(874\) −38744.8 −1.49950
\(875\) −10242.9 −0.395741
\(876\) 34572.3 1.33344
\(877\) 40861.3 1.57330 0.786652 0.617397i \(-0.211813\pi\)
0.786652 + 0.617397i \(0.211813\pi\)
\(878\) 77073.9 2.96255
\(879\) −6737.76 −0.258543
\(880\) 62476.6 2.39328
\(881\) −43839.0 −1.67647 −0.838236 0.545308i \(-0.816413\pi\)
−0.838236 + 0.545308i \(0.816413\pi\)
\(882\) −2326.24 −0.0888079
\(883\) 44625.1 1.70074 0.850371 0.526183i \(-0.176377\pi\)
0.850371 + 0.526183i \(0.176377\pi\)
\(884\) 7801.01 0.296806
\(885\) −1135.91 −0.0431450
\(886\) 80388.6 3.04820
\(887\) −43967.5 −1.66436 −0.832178 0.554509i \(-0.812906\pi\)
−0.832178 + 0.554509i \(0.812906\pi\)
\(888\) 48632.9 1.83785
\(889\) −12004.2 −0.452878
\(890\) 19646.4 0.739943
\(891\) 2814.68 0.105831
\(892\) −74933.5 −2.81273
\(893\) −14690.2 −0.550491
\(894\) −11771.6 −0.440380
\(895\) −29588.9 −1.10508
\(896\) −5162.95 −0.192502
\(897\) 14026.0 0.522089
\(898\) −56329.7 −2.09326
\(899\) −10330.1 −0.383234
\(900\) −2444.57 −0.0905396
\(901\) 5664.21 0.209436
\(902\) 62000.4 2.28868
\(903\) 142.883 0.00526563
\(904\) −56547.4 −2.08046
\(905\) 32768.2 1.20359
\(906\) −9601.89 −0.352099
\(907\) −13584.3 −0.497309 −0.248654 0.968592i \(-0.579988\pi\)
−0.248654 + 0.968592i \(0.579988\pi\)
\(908\) 36232.1 1.32423
\(909\) 4273.67 0.155939
\(910\) 14529.7 0.529291
\(911\) −16421.6 −0.597226 −0.298613 0.954374i \(-0.596524\pi\)
−0.298613 + 0.954374i \(0.596524\pi\)
\(912\) −29959.3 −1.08778
\(913\) 46369.0 1.68082
\(914\) −22303.6 −0.807152
\(915\) 1829.15 0.0660873
\(916\) −16856.0 −0.608010
\(917\) 3294.28 0.118633
\(918\) 1502.54 0.0540208
\(919\) −29487.3 −1.05843 −0.529214 0.848488i \(-0.677513\pi\)
−0.529214 + 0.848488i \(0.677513\pi\)
\(920\) 82483.7 2.95588
\(921\) −9591.23 −0.343151
\(922\) −4800.81 −0.171482
\(923\) −13567.6 −0.483839
\(924\) 14466.8 0.515067
\(925\) 3560.85 0.126573
\(926\) −23505.9 −0.834181
\(927\) −17996.3 −0.637622
\(928\) 14159.7 0.500878
\(929\) 3441.85 0.121554 0.0607769 0.998151i \(-0.480642\pi\)
0.0607769 + 0.998151i \(0.480642\pi\)
\(930\) −48714.6 −1.71765
\(931\) −2871.30 −0.101077
\(932\) −130667. −4.59242
\(933\) −10066.8 −0.353240
\(934\) 23364.8 0.818545
\(935\) −3867.55 −0.135275
\(936\) 20938.6 0.731196
\(937\) 5646.60 0.196869 0.0984346 0.995144i \(-0.468616\pi\)
0.0984346 + 0.995144i \(0.468616\pi\)
\(938\) −17786.2 −0.619125
\(939\) −6769.18 −0.235254
\(940\) 52432.2 1.81931
\(941\) −44680.1 −1.54785 −0.773927 0.633275i \(-0.781710\pi\)
−0.773927 + 0.633275i \(0.781710\pi\)
\(942\) −49290.7 −1.70486
\(943\) 42398.4 1.46414
\(944\) −6116.54 −0.210886
\(945\) 1993.92 0.0686372
\(946\) −1247.16 −0.0428633
\(947\) −48924.6 −1.67881 −0.839406 0.543505i \(-0.817097\pi\)
−0.839406 + 0.543505i \(0.817097\pi\)
\(948\) −41256.8 −1.41346
\(949\) −21681.9 −0.741647
\(950\) −4234.97 −0.144632
\(951\) −18417.6 −0.628002
\(952\) 4606.29 0.156818
\(953\) 52014.3 1.76801 0.884003 0.467482i \(-0.154839\pi\)
0.884003 + 0.467482i \(0.154839\pi\)
\(954\) 25488.9 0.865026
\(955\) 2763.55 0.0936401
\(956\) −3619.14 −0.122439
\(957\) −3690.57 −0.124660
\(958\) −14521.1 −0.489723
\(959\) −3107.37 −0.104632
\(960\) 23623.9 0.794228
\(961\) 55352.8 1.85804
\(962\) −51134.5 −1.71377
\(963\) 10500.6 0.351379
\(964\) 30210.9 1.00936
\(965\) 42738.2 1.42569
\(966\) 13885.1 0.462470
\(967\) −47117.7 −1.56691 −0.783456 0.621448i \(-0.786545\pi\)
−0.783456 + 0.621448i \(0.786545\pi\)
\(968\) 7702.95 0.255767
\(969\) 1854.60 0.0614843
\(970\) −80462.3 −2.66339
\(971\) −8195.04 −0.270846 −0.135423 0.990788i \(-0.543239\pi\)
−0.135423 + 0.990788i \(0.543239\pi\)
\(972\) 4817.41 0.158970
\(973\) 11689.9 0.385159
\(974\) 3537.35 0.116370
\(975\) 1533.10 0.0503574
\(976\) 9849.42 0.323025
\(977\) 4643.51 0.152056 0.0760282 0.997106i \(-0.475776\pi\)
0.0760282 + 0.997106i \(0.475776\pi\)
\(978\) −38186.3 −1.24853
\(979\) −12267.8 −0.400490
\(980\) 10248.2 0.334049
\(981\) −12034.6 −0.391677
\(982\) 43490.1 1.41326
\(983\) 43986.5 1.42721 0.713607 0.700546i \(-0.247060\pi\)
0.713607 + 0.700546i \(0.247060\pi\)
\(984\) 63294.1 2.05055
\(985\) −30329.0 −0.981077
\(986\) −1970.10 −0.0636317
\(987\) 5264.58 0.169781
\(988\) 43329.9 1.39525
\(989\) −852.859 −0.0274210
\(990\) −17404.0 −0.558722
\(991\) 1595.21 0.0511337 0.0255668 0.999673i \(-0.491861\pi\)
0.0255668 + 0.999673i \(0.491861\pi\)
\(992\) −116709. −3.73539
\(993\) 21089.4 0.673971
\(994\) −13431.3 −0.428588
\(995\) −32356.0 −1.03091
\(996\) 79362.0 2.52478
\(997\) 21501.2 0.682998 0.341499 0.939882i \(-0.389065\pi\)
0.341499 + 0.939882i \(0.389065\pi\)
\(998\) −43069.2 −1.36606
\(999\) −7017.22 −0.222237
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 21.4.a.c.1.1 2
3.2 odd 2 63.4.a.e.1.2 2
4.3 odd 2 336.4.a.m.1.2 2
5.2 odd 4 525.4.d.g.274.1 4
5.3 odd 4 525.4.d.g.274.4 4
5.4 even 2 525.4.a.n.1.2 2
7.2 even 3 147.4.e.l.67.2 4
7.3 odd 6 147.4.e.m.79.2 4
7.4 even 3 147.4.e.l.79.2 4
7.5 odd 6 147.4.e.m.67.2 4
7.6 odd 2 147.4.a.i.1.1 2
8.3 odd 2 1344.4.a.bo.1.1 2
8.5 even 2 1344.4.a.bg.1.1 2
12.11 even 2 1008.4.a.ba.1.1 2
15.14 odd 2 1575.4.a.p.1.1 2
21.2 odd 6 441.4.e.q.361.1 4
21.5 even 6 441.4.e.p.361.1 4
21.11 odd 6 441.4.e.q.226.1 4
21.17 even 6 441.4.e.p.226.1 4
21.20 even 2 441.4.a.r.1.2 2
28.27 even 2 2352.4.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.1 2 1.1 even 1 trivial
63.4.a.e.1.2 2 3.2 odd 2
147.4.a.i.1.1 2 7.6 odd 2
147.4.e.l.67.2 4 7.2 even 3
147.4.e.l.79.2 4 7.4 even 3
147.4.e.m.67.2 4 7.5 odd 6
147.4.e.m.79.2 4 7.3 odd 6
336.4.a.m.1.2 2 4.3 odd 2
441.4.a.r.1.2 2 21.20 even 2
441.4.e.p.226.1 4 21.17 even 6
441.4.e.p.361.1 4 21.5 even 6
441.4.e.q.226.1 4 21.11 odd 6
441.4.e.q.361.1 4 21.2 odd 6
525.4.a.n.1.2 2 5.4 even 2
525.4.d.g.274.1 4 5.2 odd 4
525.4.d.g.274.4 4 5.3 odd 4
1008.4.a.ba.1.1 2 12.11 even 2
1344.4.a.bg.1.1 2 8.5 even 2
1344.4.a.bo.1.1 2 8.3 odd 2
1575.4.a.p.1.1 2 15.14 odd 2
2352.4.a.bz.1.1 2 28.27 even 2