Properties

Label 6003.2.a.m.1.10
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} - 93 x^{2} - 369 x - 108\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.96728\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.96728 q^{2} +1.87020 q^{4} +3.90206 q^{5} -0.839519 q^{7} -0.255353 q^{8} +O(q^{10})\) \(q+1.96728 q^{2} +1.87020 q^{4} +3.90206 q^{5} -0.839519 q^{7} -0.255353 q^{8} +7.67646 q^{10} -4.08430 q^{11} -6.57808 q^{13} -1.65157 q^{14} -4.24275 q^{16} -0.379112 q^{17} -6.25665 q^{19} +7.29764 q^{20} -8.03497 q^{22} -1.00000 q^{23} +10.2261 q^{25} -12.9409 q^{26} -1.57007 q^{28} +1.00000 q^{29} +3.23356 q^{31} -7.83599 q^{32} -0.745820 q^{34} -3.27585 q^{35} -7.47652 q^{37} -12.3086 q^{38} -0.996402 q^{40} +9.84358 q^{41} -6.99632 q^{43} -7.63846 q^{44} -1.96728 q^{46} +9.65140 q^{47} -6.29521 q^{49} +20.1176 q^{50} -12.3023 q^{52} -10.9060 q^{53} -15.9372 q^{55} +0.214373 q^{56} +1.96728 q^{58} +11.3965 q^{59} +2.37385 q^{61} +6.36133 q^{62} -6.93009 q^{64} -25.6681 q^{65} -11.1611 q^{67} -0.709015 q^{68} -6.44453 q^{70} -4.24762 q^{71} -8.99243 q^{73} -14.7084 q^{74} -11.7012 q^{76} +3.42884 q^{77} +1.79119 q^{79} -16.5555 q^{80} +19.3651 q^{82} +6.12986 q^{83} -1.47932 q^{85} -13.7637 q^{86} +1.04294 q^{88} -4.65351 q^{89} +5.52242 q^{91} -1.87020 q^{92} +18.9870 q^{94} -24.4138 q^{95} -2.57912 q^{97} -12.3845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 2q^{2} + 18q^{4} - 2q^{5} + 3q^{7} - 18q^{8} + O(q^{10}) \) \( 11q - 2q^{2} + 18q^{4} - 2q^{5} + 3q^{7} - 18q^{8} + 14q^{10} - 11q^{11} - 5q^{13} - 17q^{14} + 20q^{16} - 15q^{17} - 6q^{19} - 21q^{20} - 10q^{22} - 11q^{23} + 3q^{25} + 5q^{26} + 7q^{28} + 11q^{29} + 35q^{31} - 28q^{32} + 28q^{34} - 15q^{35} - 28q^{37} + 2q^{38} - q^{40} - 10q^{41} - 6q^{43} - 18q^{44} + 2q^{46} - 15q^{47} + 22q^{49} - 15q^{50} - 36q^{52} + 7q^{53} - 12q^{55} - 56q^{56} - 2q^{58} + 20q^{59} - 20q^{61} + 11q^{62} + 36q^{64} - 11q^{65} - 39q^{67} - 35q^{68} + 38q^{70} - 49q^{71} - 3q^{73} - 37q^{74} - 18q^{76} - 25q^{77} + 41q^{79} - 51q^{80} - 19q^{82} - 13q^{83} - 62q^{86} - 40q^{88} - 34q^{89} + 2q^{91} - 18q^{92} - 14q^{94} - 25q^{95} - 11q^{97} - 53q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.96728 1.39108 0.695539 0.718488i \(-0.255165\pi\)
0.695539 + 0.718488i \(0.255165\pi\)
\(3\) 0 0
\(4\) 1.87020 0.935100
\(5\) 3.90206 1.74505 0.872527 0.488565i \(-0.162480\pi\)
0.872527 + 0.488565i \(0.162480\pi\)
\(6\) 0 0
\(7\) −0.839519 −0.317308 −0.158654 0.987334i \(-0.550715\pi\)
−0.158654 + 0.987334i \(0.550715\pi\)
\(8\) −0.255353 −0.0902808
\(9\) 0 0
\(10\) 7.67646 2.42751
\(11\) −4.08430 −1.23146 −0.615731 0.787956i \(-0.711139\pi\)
−0.615731 + 0.787956i \(0.711139\pi\)
\(12\) 0 0
\(13\) −6.57808 −1.82443 −0.912215 0.409712i \(-0.865629\pi\)
−0.912215 + 0.409712i \(0.865629\pi\)
\(14\) −1.65157 −0.441401
\(15\) 0 0
\(16\) −4.24275 −1.06069
\(17\) −0.379112 −0.0919481 −0.0459741 0.998943i \(-0.514639\pi\)
−0.0459741 + 0.998943i \(0.514639\pi\)
\(18\) 0 0
\(19\) −6.25665 −1.43537 −0.717687 0.696366i \(-0.754799\pi\)
−0.717687 + 0.696366i \(0.754799\pi\)
\(20\) 7.29764 1.63180
\(21\) 0 0
\(22\) −8.03497 −1.71306
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 10.2261 2.04522
\(26\) −12.9409 −2.53793
\(27\) 0 0
\(28\) −1.57007 −0.296715
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 3.23356 0.580765 0.290383 0.956911i \(-0.406217\pi\)
0.290383 + 0.956911i \(0.406217\pi\)
\(32\) −7.83599 −1.38522
\(33\) 0 0
\(34\) −0.745820 −0.127907
\(35\) −3.27585 −0.553720
\(36\) 0 0
\(37\) −7.47652 −1.22913 −0.614566 0.788865i \(-0.710669\pi\)
−0.614566 + 0.788865i \(0.710669\pi\)
\(38\) −12.3086 −1.99672
\(39\) 0 0
\(40\) −0.996402 −0.157545
\(41\) 9.84358 1.53731 0.768655 0.639664i \(-0.220927\pi\)
0.768655 + 0.639664i \(0.220927\pi\)
\(42\) 0 0
\(43\) −6.99632 −1.06693 −0.533465 0.845822i \(-0.679110\pi\)
−0.533465 + 0.845822i \(0.679110\pi\)
\(44\) −7.63846 −1.15154
\(45\) 0 0
\(46\) −1.96728 −0.290060
\(47\) 9.65140 1.40780 0.703900 0.710299i \(-0.251440\pi\)
0.703900 + 0.710299i \(0.251440\pi\)
\(48\) 0 0
\(49\) −6.29521 −0.899316
\(50\) 20.1176 2.84506
\(51\) 0 0
\(52\) −12.3023 −1.70602
\(53\) −10.9060 −1.49805 −0.749027 0.662540i \(-0.769479\pi\)
−0.749027 + 0.662540i \(0.769479\pi\)
\(54\) 0 0
\(55\) −15.9372 −2.14897
\(56\) 0.214373 0.0286468
\(57\) 0 0
\(58\) 1.96728 0.258317
\(59\) 11.3965 1.48370 0.741850 0.670566i \(-0.233949\pi\)
0.741850 + 0.670566i \(0.233949\pi\)
\(60\) 0 0
\(61\) 2.37385 0.303940 0.151970 0.988385i \(-0.451438\pi\)
0.151970 + 0.988385i \(0.451438\pi\)
\(62\) 6.36133 0.807890
\(63\) 0 0
\(64\) −6.93009 −0.866262
\(65\) −25.6681 −3.18373
\(66\) 0 0
\(67\) −11.1611 −1.36355 −0.681774 0.731563i \(-0.738791\pi\)
−0.681774 + 0.731563i \(0.738791\pi\)
\(68\) −0.709015 −0.0859807
\(69\) 0 0
\(70\) −6.44453 −0.770268
\(71\) −4.24762 −0.504100 −0.252050 0.967714i \(-0.581105\pi\)
−0.252050 + 0.967714i \(0.581105\pi\)
\(72\) 0 0
\(73\) −8.99243 −1.05248 −0.526242 0.850335i \(-0.676400\pi\)
−0.526242 + 0.850335i \(0.676400\pi\)
\(74\) −14.7084 −1.70982
\(75\) 0 0
\(76\) −11.7012 −1.34222
\(77\) 3.42884 0.390753
\(78\) 0 0
\(79\) 1.79119 0.201524 0.100762 0.994911i \(-0.467872\pi\)
0.100762 + 0.994911i \(0.467872\pi\)
\(80\) −16.5555 −1.85096
\(81\) 0 0
\(82\) 19.3651 2.13852
\(83\) 6.12986 0.672839 0.336420 0.941712i \(-0.390784\pi\)
0.336420 + 0.941712i \(0.390784\pi\)
\(84\) 0 0
\(85\) −1.47932 −0.160455
\(86\) −13.7637 −1.48418
\(87\) 0 0
\(88\) 1.04294 0.111177
\(89\) −4.65351 −0.493271 −0.246636 0.969108i \(-0.579325\pi\)
−0.246636 + 0.969108i \(0.579325\pi\)
\(90\) 0 0
\(91\) 5.52242 0.578907
\(92\) −1.87020 −0.194982
\(93\) 0 0
\(94\) 18.9870 1.95836
\(95\) −24.4138 −2.50480
\(96\) 0 0
\(97\) −2.57912 −0.261870 −0.130935 0.991391i \(-0.541798\pi\)
−0.130935 + 0.991391i \(0.541798\pi\)
\(98\) −12.3845 −1.25102
\(99\) 0 0
\(100\) 19.1248 1.91248
\(101\) −17.5570 −1.74699 −0.873495 0.486834i \(-0.838152\pi\)
−0.873495 + 0.486834i \(0.838152\pi\)
\(102\) 0 0
\(103\) 10.5650 1.04100 0.520500 0.853862i \(-0.325746\pi\)
0.520500 + 0.853862i \(0.325746\pi\)
\(104\) 1.67973 0.164711
\(105\) 0 0
\(106\) −21.4552 −2.08391
\(107\) 10.3172 0.997405 0.498703 0.866773i \(-0.333810\pi\)
0.498703 + 0.866773i \(0.333810\pi\)
\(108\) 0 0
\(109\) 5.06258 0.484907 0.242454 0.970163i \(-0.422048\pi\)
0.242454 + 0.970163i \(0.422048\pi\)
\(110\) −31.3529 −2.98939
\(111\) 0 0
\(112\) 3.56187 0.336565
\(113\) −3.10215 −0.291826 −0.145913 0.989297i \(-0.546612\pi\)
−0.145913 + 0.989297i \(0.546612\pi\)
\(114\) 0 0
\(115\) −3.90206 −0.363869
\(116\) 1.87020 0.173644
\(117\) 0 0
\(118\) 22.4202 2.06394
\(119\) 0.318271 0.0291759
\(120\) 0 0
\(121\) 5.68150 0.516500
\(122\) 4.67003 0.422805
\(123\) 0 0
\(124\) 6.04741 0.543074
\(125\) 20.3925 1.82396
\(126\) 0 0
\(127\) −10.0701 −0.893578 −0.446789 0.894639i \(-0.647433\pi\)
−0.446789 + 0.894639i \(0.647433\pi\)
\(128\) 2.03852 0.180181
\(129\) 0 0
\(130\) −50.4963 −4.42882
\(131\) 16.4759 1.43950 0.719752 0.694231i \(-0.244256\pi\)
0.719752 + 0.694231i \(0.244256\pi\)
\(132\) 0 0
\(133\) 5.25257 0.455456
\(134\) −21.9571 −1.89680
\(135\) 0 0
\(136\) 0.0968072 0.00830115
\(137\) −19.6782 −1.68122 −0.840610 0.541640i \(-0.817804\pi\)
−0.840610 + 0.541640i \(0.817804\pi\)
\(138\) 0 0
\(139\) 19.0807 1.61840 0.809202 0.587530i \(-0.199900\pi\)
0.809202 + 0.587530i \(0.199900\pi\)
\(140\) −6.12650 −0.517784
\(141\) 0 0
\(142\) −8.35628 −0.701243
\(143\) 26.8668 2.24672
\(144\) 0 0
\(145\) 3.90206 0.324049
\(146\) −17.6907 −1.46409
\(147\) 0 0
\(148\) −13.9826 −1.14936
\(149\) −0.870886 −0.0713457 −0.0356729 0.999364i \(-0.511357\pi\)
−0.0356729 + 0.999364i \(0.511357\pi\)
\(150\) 0 0
\(151\) 11.5494 0.939873 0.469937 0.882700i \(-0.344277\pi\)
0.469937 + 0.882700i \(0.344277\pi\)
\(152\) 1.59765 0.129587
\(153\) 0 0
\(154\) 6.74551 0.543568
\(155\) 12.6176 1.01347
\(156\) 0 0
\(157\) −6.51319 −0.519810 −0.259905 0.965634i \(-0.583691\pi\)
−0.259905 + 0.965634i \(0.583691\pi\)
\(158\) 3.52377 0.280336
\(159\) 0 0
\(160\) −30.5765 −2.41728
\(161\) 0.839519 0.0661633
\(162\) 0 0
\(163\) 1.58258 0.123957 0.0619786 0.998077i \(-0.480259\pi\)
0.0619786 + 0.998077i \(0.480259\pi\)
\(164\) 18.4095 1.43754
\(165\) 0 0
\(166\) 12.0592 0.935972
\(167\) −6.04190 −0.467536 −0.233768 0.972292i \(-0.575106\pi\)
−0.233768 + 0.972292i \(0.575106\pi\)
\(168\) 0 0
\(169\) 30.2711 2.32854
\(170\) −2.91024 −0.223205
\(171\) 0 0
\(172\) −13.0845 −0.997686
\(173\) 2.34652 0.178403 0.0892013 0.996014i \(-0.471569\pi\)
0.0892013 + 0.996014i \(0.471569\pi\)
\(174\) 0 0
\(175\) −8.58499 −0.648964
\(176\) 17.3287 1.30620
\(177\) 0 0
\(178\) −9.15477 −0.686179
\(179\) −8.16175 −0.610038 −0.305019 0.952346i \(-0.598663\pi\)
−0.305019 + 0.952346i \(0.598663\pi\)
\(180\) 0 0
\(181\) −3.58883 −0.266756 −0.133378 0.991065i \(-0.542582\pi\)
−0.133378 + 0.991065i \(0.542582\pi\)
\(182\) 10.8642 0.805305
\(183\) 0 0
\(184\) 0.255353 0.0188248
\(185\) −29.1738 −2.14490
\(186\) 0 0
\(187\) 1.54841 0.113231
\(188\) 18.0500 1.31643
\(189\) 0 0
\(190\) −48.0289 −3.48438
\(191\) −25.9036 −1.87432 −0.937159 0.348902i \(-0.886555\pi\)
−0.937159 + 0.348902i \(0.886555\pi\)
\(192\) 0 0
\(193\) −12.9762 −0.934050 −0.467025 0.884244i \(-0.654674\pi\)
−0.467025 + 0.884244i \(0.654674\pi\)
\(194\) −5.07386 −0.364282
\(195\) 0 0
\(196\) −11.7733 −0.840950
\(197\) 10.6793 0.760867 0.380433 0.924808i \(-0.375775\pi\)
0.380433 + 0.924808i \(0.375775\pi\)
\(198\) 0 0
\(199\) 3.77912 0.267895 0.133947 0.990988i \(-0.457235\pi\)
0.133947 + 0.990988i \(0.457235\pi\)
\(200\) −2.61126 −0.184644
\(201\) 0 0
\(202\) −34.5396 −2.43020
\(203\) −0.839519 −0.0589226
\(204\) 0 0
\(205\) 38.4103 2.68269
\(206\) 20.7843 1.44811
\(207\) 0 0
\(208\) 27.9091 1.93515
\(209\) 25.5540 1.76761
\(210\) 0 0
\(211\) 19.8083 1.36366 0.681829 0.731512i \(-0.261185\pi\)
0.681829 + 0.731512i \(0.261185\pi\)
\(212\) −20.3964 −1.40083
\(213\) 0 0
\(214\) 20.2969 1.38747
\(215\) −27.3001 −1.86185
\(216\) 0 0
\(217\) −2.71464 −0.184282
\(218\) 9.95953 0.674544
\(219\) 0 0
\(220\) −29.8057 −2.00950
\(221\) 2.49383 0.167753
\(222\) 0 0
\(223\) 21.9181 1.46775 0.733874 0.679286i \(-0.237710\pi\)
0.733874 + 0.679286i \(0.237710\pi\)
\(224\) 6.57845 0.439541
\(225\) 0 0
\(226\) −6.10281 −0.405953
\(227\) 12.5521 0.833113 0.416556 0.909110i \(-0.363237\pi\)
0.416556 + 0.909110i \(0.363237\pi\)
\(228\) 0 0
\(229\) 4.66018 0.307953 0.153977 0.988074i \(-0.450792\pi\)
0.153977 + 0.988074i \(0.450792\pi\)
\(230\) −7.67646 −0.506171
\(231\) 0 0
\(232\) −0.255353 −0.0167647
\(233\) −13.1954 −0.864459 −0.432229 0.901764i \(-0.642273\pi\)
−0.432229 + 0.901764i \(0.642273\pi\)
\(234\) 0 0
\(235\) 37.6603 2.45669
\(236\) 21.3138 1.38741
\(237\) 0 0
\(238\) 0.626130 0.0405860
\(239\) 4.52134 0.292461 0.146231 0.989251i \(-0.453286\pi\)
0.146231 + 0.989251i \(0.453286\pi\)
\(240\) 0 0
\(241\) 0.464697 0.0299338 0.0149669 0.999888i \(-0.495236\pi\)
0.0149669 + 0.999888i \(0.495236\pi\)
\(242\) 11.1771 0.718492
\(243\) 0 0
\(244\) 4.43958 0.284215
\(245\) −24.5643 −1.56936
\(246\) 0 0
\(247\) 41.1567 2.61874
\(248\) −0.825699 −0.0524319
\(249\) 0 0
\(250\) 40.1178 2.53727
\(251\) 8.29616 0.523649 0.261825 0.965115i \(-0.415676\pi\)
0.261825 + 0.965115i \(0.415676\pi\)
\(252\) 0 0
\(253\) 4.08430 0.256778
\(254\) −19.8108 −1.24304
\(255\) 0 0
\(256\) 17.8705 1.11691
\(257\) 9.38027 0.585125 0.292563 0.956246i \(-0.405492\pi\)
0.292563 + 0.956246i \(0.405492\pi\)
\(258\) 0 0
\(259\) 6.27668 0.390014
\(260\) −48.0044 −2.97711
\(261\) 0 0
\(262\) 32.4127 2.00246
\(263\) 19.7330 1.21679 0.608394 0.793635i \(-0.291814\pi\)
0.608394 + 0.793635i \(0.291814\pi\)
\(264\) 0 0
\(265\) −42.5559 −2.61419
\(266\) 10.3333 0.633575
\(267\) 0 0
\(268\) −20.8735 −1.27505
\(269\) −23.5286 −1.43457 −0.717283 0.696782i \(-0.754615\pi\)
−0.717283 + 0.696782i \(0.754615\pi\)
\(270\) 0 0
\(271\) −9.85432 −0.598607 −0.299304 0.954158i \(-0.596754\pi\)
−0.299304 + 0.954158i \(0.596754\pi\)
\(272\) 1.60848 0.0975282
\(273\) 0 0
\(274\) −38.7126 −2.33871
\(275\) −41.7664 −2.51861
\(276\) 0 0
\(277\) −27.2188 −1.63542 −0.817709 0.575632i \(-0.804756\pi\)
−0.817709 + 0.575632i \(0.804756\pi\)
\(278\) 37.5371 2.25133
\(279\) 0 0
\(280\) 0.836498 0.0499903
\(281\) 19.1639 1.14322 0.571611 0.820525i \(-0.306319\pi\)
0.571611 + 0.820525i \(0.306319\pi\)
\(282\) 0 0
\(283\) −18.5391 −1.10203 −0.551017 0.834494i \(-0.685760\pi\)
−0.551017 + 0.834494i \(0.685760\pi\)
\(284\) −7.94391 −0.471384
\(285\) 0 0
\(286\) 52.8546 3.12536
\(287\) −8.26387 −0.487801
\(288\) 0 0
\(289\) −16.8563 −0.991546
\(290\) 7.67646 0.450777
\(291\) 0 0
\(292\) −16.8177 −0.984179
\(293\) −13.4465 −0.785550 −0.392775 0.919635i \(-0.628485\pi\)
−0.392775 + 0.919635i \(0.628485\pi\)
\(294\) 0 0
\(295\) 44.4699 2.58914
\(296\) 1.90915 0.110967
\(297\) 0 0
\(298\) −1.71328 −0.0992476
\(299\) 6.57808 0.380420
\(300\) 0 0
\(301\) 5.87354 0.338545
\(302\) 22.7208 1.30744
\(303\) 0 0
\(304\) 26.5454 1.52248
\(305\) 9.26291 0.530393
\(306\) 0 0
\(307\) 20.2033 1.15306 0.576532 0.817074i \(-0.304406\pi\)
0.576532 + 0.817074i \(0.304406\pi\)
\(308\) 6.41263 0.365393
\(309\) 0 0
\(310\) 24.8223 1.40981
\(311\) 7.95378 0.451017 0.225509 0.974241i \(-0.427596\pi\)
0.225509 + 0.974241i \(0.427596\pi\)
\(312\) 0 0
\(313\) 5.26012 0.297319 0.148660 0.988888i \(-0.452504\pi\)
0.148660 + 0.988888i \(0.452504\pi\)
\(314\) −12.8133 −0.723096
\(315\) 0 0
\(316\) 3.34988 0.188446
\(317\) −32.0143 −1.79810 −0.899052 0.437841i \(-0.855743\pi\)
−0.899052 + 0.437841i \(0.855743\pi\)
\(318\) 0 0
\(319\) −4.08430 −0.228677
\(320\) −27.0417 −1.51167
\(321\) 0 0
\(322\) 1.65157 0.0920384
\(323\) 2.37197 0.131980
\(324\) 0 0
\(325\) −67.2680 −3.73135
\(326\) 3.11338 0.172434
\(327\) 0 0
\(328\) −2.51358 −0.138789
\(329\) −8.10253 −0.446707
\(330\) 0 0
\(331\) −27.8202 −1.52914 −0.764569 0.644542i \(-0.777048\pi\)
−0.764569 + 0.644542i \(0.777048\pi\)
\(332\) 11.4641 0.629172
\(333\) 0 0
\(334\) −11.8861 −0.650380
\(335\) −43.5514 −2.37947
\(336\) 0 0
\(337\) −20.1838 −1.09948 −0.549742 0.835334i \(-0.685274\pi\)
−0.549742 + 0.835334i \(0.685274\pi\)
\(338\) 59.5518 3.23919
\(339\) 0 0
\(340\) −2.76662 −0.150041
\(341\) −13.2068 −0.715190
\(342\) 0 0
\(343\) 11.1616 0.602668
\(344\) 1.78653 0.0963232
\(345\) 0 0
\(346\) 4.61627 0.248172
\(347\) −1.48365 −0.0796466 −0.0398233 0.999207i \(-0.512680\pi\)
−0.0398233 + 0.999207i \(0.512680\pi\)
\(348\) 0 0
\(349\) 12.5241 0.670398 0.335199 0.942147i \(-0.391197\pi\)
0.335199 + 0.942147i \(0.391197\pi\)
\(350\) −16.8891 −0.902760
\(351\) 0 0
\(352\) 32.0045 1.70585
\(353\) −22.5177 −1.19850 −0.599248 0.800564i \(-0.704534\pi\)
−0.599248 + 0.800564i \(0.704534\pi\)
\(354\) 0 0
\(355\) −16.5745 −0.879683
\(356\) −8.70300 −0.461258
\(357\) 0 0
\(358\) −16.0565 −0.848610
\(359\) 31.5736 1.66639 0.833196 0.552978i \(-0.186508\pi\)
0.833196 + 0.552978i \(0.186508\pi\)
\(360\) 0 0
\(361\) 20.1456 1.06030
\(362\) −7.06024 −0.371078
\(363\) 0 0
\(364\) 10.3280 0.541336
\(365\) −35.0890 −1.83664
\(366\) 0 0
\(367\) 13.4332 0.701207 0.350604 0.936524i \(-0.385976\pi\)
0.350604 + 0.936524i \(0.385976\pi\)
\(368\) 4.24275 0.221169
\(369\) 0 0
\(370\) −57.3932 −2.98373
\(371\) 9.15578 0.475345
\(372\) 0 0
\(373\) 6.88479 0.356481 0.178240 0.983987i \(-0.442960\pi\)
0.178240 + 0.983987i \(0.442960\pi\)
\(374\) 3.04615 0.157513
\(375\) 0 0
\(376\) −2.46451 −0.127097
\(377\) −6.57808 −0.338788
\(378\) 0 0
\(379\) 3.53749 0.181708 0.0908542 0.995864i \(-0.471040\pi\)
0.0908542 + 0.995864i \(0.471040\pi\)
\(380\) −45.6587 −2.34224
\(381\) 0 0
\(382\) −50.9597 −2.60732
\(383\) −34.7172 −1.77396 −0.886982 0.461804i \(-0.847202\pi\)
−0.886982 + 0.461804i \(0.847202\pi\)
\(384\) 0 0
\(385\) 13.3796 0.681886
\(386\) −25.5279 −1.29934
\(387\) 0 0
\(388\) −4.82348 −0.244875
\(389\) 34.0537 1.72659 0.863294 0.504701i \(-0.168397\pi\)
0.863294 + 0.504701i \(0.168397\pi\)
\(390\) 0 0
\(391\) 0.379112 0.0191725
\(392\) 1.60750 0.0811909
\(393\) 0 0
\(394\) 21.0092 1.05843
\(395\) 6.98933 0.351671
\(396\) 0 0
\(397\) −18.1742 −0.912135 −0.456068 0.889945i \(-0.650743\pi\)
−0.456068 + 0.889945i \(0.650743\pi\)
\(398\) 7.43460 0.372663
\(399\) 0 0
\(400\) −43.3867 −2.16934
\(401\) −8.34533 −0.416746 −0.208373 0.978049i \(-0.566817\pi\)
−0.208373 + 0.978049i \(0.566817\pi\)
\(402\) 0 0
\(403\) −21.2706 −1.05957
\(404\) −32.8352 −1.63361
\(405\) 0 0
\(406\) −1.65157 −0.0819660
\(407\) 30.5363 1.51363
\(408\) 0 0
\(409\) 24.6842 1.22056 0.610278 0.792188i \(-0.291058\pi\)
0.610278 + 0.792188i \(0.291058\pi\)
\(410\) 75.5638 3.73183
\(411\) 0 0
\(412\) 19.7586 0.973439
\(413\) −9.56758 −0.470790
\(414\) 0 0
\(415\) 23.9191 1.17414
\(416\) 51.5457 2.52724
\(417\) 0 0
\(418\) 50.2720 2.45888
\(419\) 0.233296 0.0113972 0.00569862 0.999984i \(-0.498186\pi\)
0.00569862 + 0.999984i \(0.498186\pi\)
\(420\) 0 0
\(421\) 15.0075 0.731421 0.365710 0.930729i \(-0.380826\pi\)
0.365710 + 0.930729i \(0.380826\pi\)
\(422\) 38.9685 1.89696
\(423\) 0 0
\(424\) 2.78487 0.135245
\(425\) −3.87683 −0.188054
\(426\) 0 0
\(427\) −1.99289 −0.0964428
\(428\) 19.2953 0.932674
\(429\) 0 0
\(430\) −53.7070 −2.58998
\(431\) −33.8478 −1.63039 −0.815195 0.579187i \(-0.803370\pi\)
−0.815195 + 0.579187i \(0.803370\pi\)
\(432\) 0 0
\(433\) −25.3136 −1.21649 −0.608247 0.793748i \(-0.708127\pi\)
−0.608247 + 0.793748i \(0.708127\pi\)
\(434\) −5.34046 −0.256350
\(435\) 0 0
\(436\) 9.46804 0.453437
\(437\) 6.25665 0.299296
\(438\) 0 0
\(439\) −1.60242 −0.0764793 −0.0382397 0.999269i \(-0.512175\pi\)
−0.0382397 + 0.999269i \(0.512175\pi\)
\(440\) 4.06960 0.194011
\(441\) 0 0
\(442\) 4.90606 0.233357
\(443\) −12.6508 −0.601055 −0.300528 0.953773i \(-0.597163\pi\)
−0.300528 + 0.953773i \(0.597163\pi\)
\(444\) 0 0
\(445\) −18.1583 −0.860785
\(446\) 43.1192 2.04175
\(447\) 0 0
\(448\) 5.81794 0.274872
\(449\) 4.38659 0.207016 0.103508 0.994629i \(-0.466993\pi\)
0.103508 + 0.994629i \(0.466993\pi\)
\(450\) 0 0
\(451\) −40.2041 −1.89314
\(452\) −5.80165 −0.272886
\(453\) 0 0
\(454\) 24.6935 1.15893
\(455\) 21.5488 1.01022
\(456\) 0 0
\(457\) −11.9937 −0.561040 −0.280520 0.959848i \(-0.590507\pi\)
−0.280520 + 0.959848i \(0.590507\pi\)
\(458\) 9.16789 0.428387
\(459\) 0 0
\(460\) −7.29764 −0.340254
\(461\) −6.56938 −0.305967 −0.152983 0.988229i \(-0.548888\pi\)
−0.152983 + 0.988229i \(0.548888\pi\)
\(462\) 0 0
\(463\) −25.2208 −1.17211 −0.586054 0.810272i \(-0.699319\pi\)
−0.586054 + 0.810272i \(0.699319\pi\)
\(464\) −4.24275 −0.196965
\(465\) 0 0
\(466\) −25.9591 −1.20253
\(467\) −18.1238 −0.838671 −0.419336 0.907831i \(-0.637737\pi\)
−0.419336 + 0.907831i \(0.637737\pi\)
\(468\) 0 0
\(469\) 9.36997 0.432665
\(470\) 74.0885 3.41745
\(471\) 0 0
\(472\) −2.91013 −0.133950
\(473\) 28.5751 1.31388
\(474\) 0 0
\(475\) −63.9810 −2.93565
\(476\) 0.595231 0.0272824
\(477\) 0 0
\(478\) 8.89475 0.406837
\(479\) 29.4997 1.34787 0.673937 0.738789i \(-0.264602\pi\)
0.673937 + 0.738789i \(0.264602\pi\)
\(480\) 0 0
\(481\) 49.1811 2.24247
\(482\) 0.914190 0.0416402
\(483\) 0 0
\(484\) 10.6255 0.482979
\(485\) −10.0639 −0.456978
\(486\) 0 0
\(487\) −1.65098 −0.0748129 −0.0374064 0.999300i \(-0.511910\pi\)
−0.0374064 + 0.999300i \(0.511910\pi\)
\(488\) −0.606169 −0.0274400
\(489\) 0 0
\(490\) −48.3249 −2.18310
\(491\) −30.3853 −1.37127 −0.685636 0.727945i \(-0.740476\pi\)
−0.685636 + 0.727945i \(0.740476\pi\)
\(492\) 0 0
\(493\) −0.379112 −0.0170743
\(494\) 80.9668 3.64287
\(495\) 0 0
\(496\) −13.7192 −0.616010
\(497\) 3.56596 0.159955
\(498\) 0 0
\(499\) −29.1755 −1.30607 −0.653037 0.757326i \(-0.726505\pi\)
−0.653037 + 0.757326i \(0.726505\pi\)
\(500\) 38.1381 1.70559
\(501\) 0 0
\(502\) 16.3209 0.728437
\(503\) −33.6484 −1.50031 −0.750155 0.661262i \(-0.770021\pi\)
−0.750155 + 0.661262i \(0.770021\pi\)
\(504\) 0 0
\(505\) −68.5086 −3.04859
\(506\) 8.03497 0.357198
\(507\) 0 0
\(508\) −18.8331 −0.835585
\(509\) 6.99105 0.309873 0.154937 0.987924i \(-0.450483\pi\)
0.154937 + 0.987924i \(0.450483\pi\)
\(510\) 0 0
\(511\) 7.54931 0.333962
\(512\) 31.0793 1.37353
\(513\) 0 0
\(514\) 18.4536 0.813955
\(515\) 41.2252 1.81660
\(516\) 0 0
\(517\) −39.4192 −1.73365
\(518\) 12.3480 0.542540
\(519\) 0 0
\(520\) 6.55441 0.287430
\(521\) −8.97472 −0.393189 −0.196595 0.980485i \(-0.562988\pi\)
−0.196595 + 0.980485i \(0.562988\pi\)
\(522\) 0 0
\(523\) −29.4473 −1.28764 −0.643819 0.765177i \(-0.722651\pi\)
−0.643819 + 0.765177i \(0.722651\pi\)
\(524\) 30.8132 1.34608
\(525\) 0 0
\(526\) 38.8203 1.69265
\(527\) −1.22588 −0.0534003
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −83.7194 −3.63654
\(531\) 0 0
\(532\) 9.82336 0.425897
\(533\) −64.7518 −2.80471
\(534\) 0 0
\(535\) 40.2585 1.74053
\(536\) 2.85002 0.123102
\(537\) 0 0
\(538\) −46.2875 −1.99560
\(539\) 25.7115 1.10747
\(540\) 0 0
\(541\) −1.59606 −0.0686201 −0.0343101 0.999411i \(-0.510923\pi\)
−0.0343101 + 0.999411i \(0.510923\pi\)
\(542\) −19.3862 −0.832710
\(543\) 0 0
\(544\) 2.97071 0.127368
\(545\) 19.7545 0.846190
\(546\) 0 0
\(547\) 13.6372 0.583086 0.291543 0.956558i \(-0.405831\pi\)
0.291543 + 0.956558i \(0.405831\pi\)
\(548\) −36.8022 −1.57211
\(549\) 0 0
\(550\) −82.1663 −3.50358
\(551\) −6.25665 −0.266542
\(552\) 0 0
\(553\) −1.50374 −0.0639454
\(554\) −53.5470 −2.27499
\(555\) 0 0
\(556\) 35.6847 1.51337
\(557\) 1.68424 0.0713636 0.0356818 0.999363i \(-0.488640\pi\)
0.0356818 + 0.999363i \(0.488640\pi\)
\(558\) 0 0
\(559\) 46.0224 1.94654
\(560\) 13.8986 0.587324
\(561\) 0 0
\(562\) 37.7008 1.59031
\(563\) −0.563021 −0.0237285 −0.0118643 0.999930i \(-0.503777\pi\)
−0.0118643 + 0.999930i \(0.503777\pi\)
\(564\) 0 0
\(565\) −12.1048 −0.509252
\(566\) −36.4716 −1.53302
\(567\) 0 0
\(568\) 1.08464 0.0455106
\(569\) 30.3716 1.27324 0.636622 0.771176i \(-0.280331\pi\)
0.636622 + 0.771176i \(0.280331\pi\)
\(570\) 0 0
\(571\) 9.24952 0.387080 0.193540 0.981092i \(-0.438003\pi\)
0.193540 + 0.981092i \(0.438003\pi\)
\(572\) 50.2464 2.10091
\(573\) 0 0
\(574\) −16.2574 −0.678569
\(575\) −10.2261 −0.426457
\(576\) 0 0
\(577\) 35.0432 1.45887 0.729433 0.684052i \(-0.239784\pi\)
0.729433 + 0.684052i \(0.239784\pi\)
\(578\) −33.1611 −1.37932
\(579\) 0 0
\(580\) 7.29764 0.303018
\(581\) −5.14613 −0.213497
\(582\) 0 0
\(583\) 44.5433 1.84480
\(584\) 2.29624 0.0950192
\(585\) 0 0
\(586\) −26.4530 −1.09276
\(587\) 5.85702 0.241745 0.120873 0.992668i \(-0.461431\pi\)
0.120873 + 0.992668i \(0.461431\pi\)
\(588\) 0 0
\(589\) −20.2313 −0.833615
\(590\) 87.4848 3.60169
\(591\) 0 0
\(592\) 31.7210 1.30373
\(593\) 13.3743 0.549217 0.274608 0.961556i \(-0.411452\pi\)
0.274608 + 0.961556i \(0.411452\pi\)
\(594\) 0 0
\(595\) 1.24191 0.0509135
\(596\) −1.62873 −0.0667154
\(597\) 0 0
\(598\) 12.9409 0.529194
\(599\) −17.1311 −0.699958 −0.349979 0.936758i \(-0.613811\pi\)
−0.349979 + 0.936758i \(0.613811\pi\)
\(600\) 0 0
\(601\) 39.8948 1.62734 0.813671 0.581325i \(-0.197466\pi\)
0.813671 + 0.581325i \(0.197466\pi\)
\(602\) 11.5549 0.470943
\(603\) 0 0
\(604\) 21.5996 0.878876
\(605\) 22.1696 0.901320
\(606\) 0 0
\(607\) 29.2826 1.18854 0.594272 0.804264i \(-0.297440\pi\)
0.594272 + 0.804264i \(0.297440\pi\)
\(608\) 49.0270 1.98831
\(609\) 0 0
\(610\) 18.2228 0.737818
\(611\) −63.4876 −2.56843
\(612\) 0 0
\(613\) 23.7708 0.960093 0.480046 0.877243i \(-0.340620\pi\)
0.480046 + 0.877243i \(0.340620\pi\)
\(614\) 39.7456 1.60400
\(615\) 0 0
\(616\) −0.875565 −0.0352775
\(617\) 12.5181 0.503958 0.251979 0.967733i \(-0.418919\pi\)
0.251979 + 0.967733i \(0.418919\pi\)
\(618\) 0 0
\(619\) −27.2765 −1.09633 −0.548167 0.836369i \(-0.684674\pi\)
−0.548167 + 0.836369i \(0.684674\pi\)
\(620\) 23.5974 0.947693
\(621\) 0 0
\(622\) 15.6473 0.627401
\(623\) 3.90671 0.156519
\(624\) 0 0
\(625\) 28.4424 1.13769
\(626\) 10.3481 0.413595
\(627\) 0 0
\(628\) −12.1810 −0.486074
\(629\) 2.83444 0.113016
\(630\) 0 0
\(631\) −17.2183 −0.685448 −0.342724 0.939436i \(-0.611350\pi\)
−0.342724 + 0.939436i \(0.611350\pi\)
\(632\) −0.457385 −0.0181938
\(633\) 0 0
\(634\) −62.9813 −2.50131
\(635\) −39.2942 −1.55934
\(636\) 0 0
\(637\) 41.4104 1.64074
\(638\) −8.03497 −0.318107
\(639\) 0 0
\(640\) 7.95443 0.314426
\(641\) −31.7850 −1.25543 −0.627715 0.778443i \(-0.716010\pi\)
−0.627715 + 0.778443i \(0.716010\pi\)
\(642\) 0 0
\(643\) 27.2635 1.07517 0.537584 0.843210i \(-0.319337\pi\)
0.537584 + 0.843210i \(0.319337\pi\)
\(644\) 1.57007 0.0618693
\(645\) 0 0
\(646\) 4.66633 0.183594
\(647\) 6.76480 0.265952 0.132976 0.991119i \(-0.457547\pi\)
0.132976 + 0.991119i \(0.457547\pi\)
\(648\) 0 0
\(649\) −46.5468 −1.82712
\(650\) −132.335 −5.19061
\(651\) 0 0
\(652\) 2.95974 0.115912
\(653\) 9.63997 0.377241 0.188621 0.982050i \(-0.439598\pi\)
0.188621 + 0.982050i \(0.439598\pi\)
\(654\) 0 0
\(655\) 64.2899 2.51201
\(656\) −41.7639 −1.63061
\(657\) 0 0
\(658\) −15.9400 −0.621404
\(659\) −37.9893 −1.47985 −0.739927 0.672687i \(-0.765140\pi\)
−0.739927 + 0.672687i \(0.765140\pi\)
\(660\) 0 0
\(661\) −11.8372 −0.460414 −0.230207 0.973142i \(-0.573940\pi\)
−0.230207 + 0.973142i \(0.573940\pi\)
\(662\) −54.7302 −2.12715
\(663\) 0 0
\(664\) −1.56528 −0.0607445
\(665\) 20.4959 0.794795
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) −11.2996 −0.437193
\(669\) 0 0
\(670\) −85.6779 −3.31003
\(671\) −9.69551 −0.374291
\(672\) 0 0
\(673\) −11.9904 −0.462196 −0.231098 0.972931i \(-0.574232\pi\)
−0.231098 + 0.972931i \(0.574232\pi\)
\(674\) −39.7073 −1.52947
\(675\) 0 0
\(676\) 56.6130 2.17742
\(677\) 36.3152 1.39571 0.697853 0.716241i \(-0.254139\pi\)
0.697853 + 0.716241i \(0.254139\pi\)
\(678\) 0 0
\(679\) 2.16522 0.0830936
\(680\) 0.377748 0.0144860
\(681\) 0 0
\(682\) −25.9816 −0.994886
\(683\) 46.4937 1.77903 0.889516 0.456904i \(-0.151042\pi\)
0.889516 + 0.456904i \(0.151042\pi\)
\(684\) 0 0
\(685\) −76.7855 −2.93382
\(686\) 21.9580 0.838359
\(687\) 0 0
\(688\) 29.6837 1.13168
\(689\) 71.7404 2.73309
\(690\) 0 0
\(691\) 19.1780 0.729564 0.364782 0.931093i \(-0.381143\pi\)
0.364782 + 0.931093i \(0.381143\pi\)
\(692\) 4.38846 0.166824
\(693\) 0 0
\(694\) −2.91876 −0.110795
\(695\) 74.4541 2.82420
\(696\) 0 0
\(697\) −3.73182 −0.141353
\(698\) 24.6384 0.932576
\(699\) 0 0
\(700\) −16.0556 −0.606846
\(701\) −16.6639 −0.629387 −0.314694 0.949193i \(-0.601902\pi\)
−0.314694 + 0.949193i \(0.601902\pi\)
\(702\) 0 0
\(703\) 46.7779 1.76426
\(704\) 28.3046 1.06677
\(705\) 0 0
\(706\) −44.2987 −1.66720
\(707\) 14.7394 0.554334
\(708\) 0 0
\(709\) −34.2240 −1.28531 −0.642655 0.766155i \(-0.722167\pi\)
−0.642655 + 0.766155i \(0.722167\pi\)
\(710\) −32.6067 −1.22371
\(711\) 0 0
\(712\) 1.18829 0.0445329
\(713\) −3.23356 −0.121098
\(714\) 0 0
\(715\) 104.836 3.92064
\(716\) −15.2641 −0.570446
\(717\) 0 0
\(718\) 62.1142 2.31808
\(719\) 19.1950 0.715851 0.357926 0.933750i \(-0.383484\pi\)
0.357926 + 0.933750i \(0.383484\pi\)
\(720\) 0 0
\(721\) −8.86951 −0.330318
\(722\) 39.6321 1.47495
\(723\) 0 0
\(724\) −6.71183 −0.249443
\(725\) 10.2261 0.379787
\(726\) 0 0
\(727\) −1.78597 −0.0662380 −0.0331190 0.999451i \(-0.510544\pi\)
−0.0331190 + 0.999451i \(0.510544\pi\)
\(728\) −1.41016 −0.0522641
\(729\) 0 0
\(730\) −69.0300 −2.55492
\(731\) 2.65239 0.0981021
\(732\) 0 0
\(733\) −12.2076 −0.450898 −0.225449 0.974255i \(-0.572385\pi\)
−0.225449 + 0.974255i \(0.572385\pi\)
\(734\) 26.4269 0.975435
\(735\) 0 0
\(736\) 7.83599 0.288838
\(737\) 45.5854 1.67916
\(738\) 0 0
\(739\) 13.4944 0.496399 0.248200 0.968709i \(-0.420161\pi\)
0.248200 + 0.968709i \(0.420161\pi\)
\(740\) −54.5609 −2.00570
\(741\) 0 0
\(742\) 18.0120 0.661242
\(743\) −29.3204 −1.07566 −0.537830 0.843053i \(-0.680756\pi\)
−0.537830 + 0.843053i \(0.680756\pi\)
\(744\) 0 0
\(745\) −3.39825 −0.124502
\(746\) 13.5443 0.495893
\(747\) 0 0
\(748\) 2.89583 0.105882
\(749\) −8.66151 −0.316485
\(750\) 0 0
\(751\) −20.9802 −0.765577 −0.382789 0.923836i \(-0.625036\pi\)
−0.382789 + 0.923836i \(0.625036\pi\)
\(752\) −40.9485 −1.49324
\(753\) 0 0
\(754\) −12.9409 −0.471281
\(755\) 45.0663 1.64013
\(756\) 0 0
\(757\) −47.6763 −1.73283 −0.866413 0.499328i \(-0.833580\pi\)
−0.866413 + 0.499328i \(0.833580\pi\)
\(758\) 6.95923 0.252771
\(759\) 0 0
\(760\) 6.23413 0.226136
\(761\) −35.1401 −1.27383 −0.636913 0.770935i \(-0.719789\pi\)
−0.636913 + 0.770935i \(0.719789\pi\)
\(762\) 0 0
\(763\) −4.25013 −0.153865
\(764\) −48.4449 −1.75268
\(765\) 0 0
\(766\) −68.2985 −2.46772
\(767\) −74.9671 −2.70691
\(768\) 0 0
\(769\) 7.10129 0.256079 0.128040 0.991769i \(-0.459132\pi\)
0.128040 + 0.991769i \(0.459132\pi\)
\(770\) 26.3214 0.948557
\(771\) 0 0
\(772\) −24.2682 −0.873430
\(773\) 20.1935 0.726311 0.363155 0.931729i \(-0.381699\pi\)
0.363155 + 0.931729i \(0.381699\pi\)
\(774\) 0 0
\(775\) 33.0667 1.18779
\(776\) 0.658586 0.0236419
\(777\) 0 0
\(778\) 66.9932 2.40182
\(779\) −61.5878 −2.20661
\(780\) 0 0
\(781\) 17.3486 0.620780
\(782\) 0.745820 0.0266705
\(783\) 0 0
\(784\) 26.7090 0.953893
\(785\) −25.4149 −0.907096
\(786\) 0 0
\(787\) 19.3983 0.691476 0.345738 0.938331i \(-0.387629\pi\)
0.345738 + 0.938331i \(0.387629\pi\)
\(788\) 19.9724 0.711487
\(789\) 0 0
\(790\) 13.7500 0.489202
\(791\) 2.60431 0.0925988
\(792\) 0 0
\(793\) −15.6154 −0.554518
\(794\) −35.7537 −1.26885
\(795\) 0 0
\(796\) 7.06772 0.250509
\(797\) 15.2083 0.538704 0.269352 0.963042i \(-0.413190\pi\)
0.269352 + 0.963042i \(0.413190\pi\)
\(798\) 0 0
\(799\) −3.65896 −0.129445
\(800\) −80.1314 −2.83307
\(801\) 0 0
\(802\) −16.4176 −0.579727
\(803\) 36.7278 1.29610
\(804\) 0 0
\(805\) 3.27585 0.115459
\(806\) −41.8453 −1.47394
\(807\) 0 0
\(808\) 4.48323 0.157720
\(809\) 25.7908 0.906755 0.453377 0.891319i \(-0.350219\pi\)
0.453377 + 0.891319i \(0.350219\pi\)
\(810\) 0 0
\(811\) −28.2524 −0.992076 −0.496038 0.868301i \(-0.665212\pi\)
−0.496038 + 0.868301i \(0.665212\pi\)
\(812\) −1.57007 −0.0550986
\(813\) 0 0
\(814\) 60.0736 2.10558
\(815\) 6.17532 0.216312
\(816\) 0 0
\(817\) 43.7735 1.53144
\(818\) 48.5608 1.69789
\(819\) 0 0
\(820\) 71.8349 2.50858
\(821\) −48.0439 −1.67674 −0.838372 0.545099i \(-0.816492\pi\)
−0.838372 + 0.545099i \(0.816492\pi\)
\(822\) 0 0
\(823\) −11.9942 −0.418092 −0.209046 0.977906i \(-0.567036\pi\)
−0.209046 + 0.977906i \(0.567036\pi\)
\(824\) −2.69780 −0.0939823
\(825\) 0 0
\(826\) −18.8221 −0.654906
\(827\) −14.4329 −0.501880 −0.250940 0.968003i \(-0.580740\pi\)
−0.250940 + 0.968003i \(0.580740\pi\)
\(828\) 0 0
\(829\) 11.6772 0.405568 0.202784 0.979224i \(-0.435001\pi\)
0.202784 + 0.979224i \(0.435001\pi\)
\(830\) 47.0556 1.63332
\(831\) 0 0
\(832\) 45.5867 1.58043
\(833\) 2.38659 0.0826904
\(834\) 0 0
\(835\) −23.5759 −0.815877
\(836\) 47.7911 1.65289
\(837\) 0 0
\(838\) 0.458958 0.0158545
\(839\) −38.6936 −1.33585 −0.667925 0.744228i \(-0.732817\pi\)
−0.667925 + 0.744228i \(0.732817\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 29.5240 1.01746
\(843\) 0 0
\(844\) 37.0454 1.27516
\(845\) 118.120 4.06344
\(846\) 0 0
\(847\) −4.76972 −0.163890
\(848\) 46.2714 1.58897
\(849\) 0 0
\(850\) −7.62682 −0.261598
\(851\) 7.47652 0.256292
\(852\) 0 0
\(853\) −4.48283 −0.153489 −0.0767446 0.997051i \(-0.524453\pi\)
−0.0767446 + 0.997051i \(0.524453\pi\)
\(854\) −3.92058 −0.134159
\(855\) 0 0
\(856\) −2.63453 −0.0900465
\(857\) −4.53684 −0.154976 −0.0774878 0.996993i \(-0.524690\pi\)
−0.0774878 + 0.996993i \(0.524690\pi\)
\(858\) 0 0
\(859\) 36.2457 1.23669 0.618343 0.785908i \(-0.287804\pi\)
0.618343 + 0.785908i \(0.287804\pi\)
\(860\) −51.0566 −1.74102
\(861\) 0 0
\(862\) −66.5881 −2.26800
\(863\) −39.8579 −1.35678 −0.678390 0.734702i \(-0.737322\pi\)
−0.678390 + 0.734702i \(0.737322\pi\)
\(864\) 0 0
\(865\) 9.15626 0.311322
\(866\) −49.7990 −1.69224
\(867\) 0 0
\(868\) −5.07691 −0.172322
\(869\) −7.31575 −0.248170
\(870\) 0 0
\(871\) 73.4187 2.48770
\(872\) −1.29274 −0.0437778
\(873\) 0 0
\(874\) 12.3086 0.416344
\(875\) −17.1199 −0.578758
\(876\) 0 0
\(877\) −49.1279 −1.65893 −0.829466 0.558557i \(-0.811355\pi\)
−0.829466 + 0.558557i \(0.811355\pi\)
\(878\) −3.15241 −0.106389
\(879\) 0 0
\(880\) 67.6175 2.27939
\(881\) 34.2531 1.15402 0.577008 0.816738i \(-0.304220\pi\)
0.577008 + 0.816738i \(0.304220\pi\)
\(882\) 0 0
\(883\) −6.47062 −0.217754 −0.108877 0.994055i \(-0.534725\pi\)
−0.108877 + 0.994055i \(0.534725\pi\)
\(884\) 4.66395 0.156866
\(885\) 0 0
\(886\) −24.8876 −0.836115
\(887\) −5.70034 −0.191399 −0.0956993 0.995410i \(-0.530509\pi\)
−0.0956993 + 0.995410i \(0.530509\pi\)
\(888\) 0 0
\(889\) 8.45405 0.283540
\(890\) −35.7225 −1.19742
\(891\) 0 0
\(892\) 40.9913 1.37249
\(893\) −60.3854 −2.02072
\(894\) 0 0
\(895\) −31.8476 −1.06455
\(896\) −1.71137 −0.0571730
\(897\) 0 0
\(898\) 8.62967 0.287976
\(899\) 3.23356 0.107845
\(900\) 0 0
\(901\) 4.13459 0.137743
\(902\) −79.0929 −2.63350
\(903\) 0 0
\(904\) 0.792143 0.0263463
\(905\) −14.0038 −0.465503
\(906\) 0 0
\(907\) −10.6632 −0.354066 −0.177033 0.984205i \(-0.556650\pi\)
−0.177033 + 0.984205i \(0.556650\pi\)
\(908\) 23.4750 0.779044
\(909\) 0 0
\(910\) 42.3926 1.40530
\(911\) −3.27255 −0.108424 −0.0542122 0.998529i \(-0.517265\pi\)
−0.0542122 + 0.998529i \(0.517265\pi\)
\(912\) 0 0
\(913\) −25.0362 −0.828576
\(914\) −23.5949 −0.780451
\(915\) 0 0
\(916\) 8.71546 0.287967
\(917\) −13.8318 −0.456766
\(918\) 0 0
\(919\) 43.3072 1.42857 0.714286 0.699854i \(-0.246752\pi\)
0.714286 + 0.699854i \(0.246752\pi\)
\(920\) 0.996402 0.0328504
\(921\) 0 0
\(922\) −12.9238 −0.425624
\(923\) 27.9412 0.919696
\(924\) 0 0
\(925\) −76.4555 −2.51384
\(926\) −49.6164 −1.63050
\(927\) 0 0
\(928\) −7.83599 −0.257229
\(929\) −6.72374 −0.220599 −0.110299 0.993898i \(-0.535181\pi\)
−0.110299 + 0.993898i \(0.535181\pi\)
\(930\) 0 0
\(931\) 39.3869 1.29085
\(932\) −24.6780 −0.808355
\(933\) 0 0
\(934\) −35.6547 −1.16666
\(935\) 6.04197 0.197594
\(936\) 0 0
\(937\) −34.4674 −1.12600 −0.563000 0.826457i \(-0.690353\pi\)
−0.563000 + 0.826457i \(0.690353\pi\)
\(938\) 18.4334 0.601871
\(939\) 0 0
\(940\) 70.4324 2.29725
\(941\) 20.1814 0.657895 0.328947 0.944348i \(-0.393306\pi\)
0.328947 + 0.944348i \(0.393306\pi\)
\(942\) 0 0
\(943\) −9.84358 −0.320551
\(944\) −48.3526 −1.57374
\(945\) 0 0
\(946\) 56.2153 1.82772
\(947\) −54.5419 −1.77237 −0.886186 0.463329i \(-0.846655\pi\)
−0.886186 + 0.463329i \(0.846655\pi\)
\(948\) 0 0
\(949\) 59.1529 1.92018
\(950\) −125.869 −4.08372
\(951\) 0 0
\(952\) −0.0812714 −0.00263402
\(953\) 40.3802 1.30804 0.654021 0.756476i \(-0.273081\pi\)
0.654021 + 0.756476i \(0.273081\pi\)
\(954\) 0 0
\(955\) −101.077 −3.27079
\(956\) 8.45581 0.273481
\(957\) 0 0
\(958\) 58.0342 1.87500
\(959\) 16.5202 0.533465
\(960\) 0 0
\(961\) −20.5441 −0.662712
\(962\) 96.7531 3.11945
\(963\) 0 0
\(964\) 0.869076 0.0279911
\(965\) −50.6341 −1.62997
\(966\) 0 0
\(967\) 31.3292 1.00748 0.503740 0.863855i \(-0.331957\pi\)
0.503740 + 0.863855i \(0.331957\pi\)
\(968\) −1.45079 −0.0466300
\(969\) 0 0
\(970\) −19.7985 −0.635692
\(971\) −13.1463 −0.421886 −0.210943 0.977498i \(-0.567654\pi\)
−0.210943 + 0.977498i \(0.567654\pi\)
\(972\) 0 0
\(973\) −16.0186 −0.513533
\(974\) −3.24794 −0.104071
\(975\) 0 0
\(976\) −10.0717 −0.322386
\(977\) 57.3837 1.83587 0.917934 0.396732i \(-0.129856\pi\)
0.917934 + 0.396732i \(0.129856\pi\)
\(978\) 0 0
\(979\) 19.0063 0.607445
\(980\) −45.9401 −1.46750
\(981\) 0 0
\(982\) −59.7765 −1.90755
\(983\) −50.1387 −1.59918 −0.799588 0.600549i \(-0.794949\pi\)
−0.799588 + 0.600549i \(0.794949\pi\)
\(984\) 0 0
\(985\) 41.6712 1.32775
\(986\) −0.745820 −0.0237517
\(987\) 0 0
\(988\) 76.9713 2.44878
\(989\) 6.99632 0.222470
\(990\) 0 0
\(991\) −3.14635 −0.0999472 −0.0499736 0.998751i \(-0.515914\pi\)
−0.0499736 + 0.998751i \(0.515914\pi\)
\(992\) −25.3382 −0.804487
\(993\) 0 0
\(994\) 7.01525 0.222510
\(995\) 14.7464 0.467491
\(996\) 0 0
\(997\) 24.3387 0.770813 0.385406 0.922747i \(-0.374061\pi\)
0.385406 + 0.922747i \(0.374061\pi\)
\(998\) −57.3964 −1.81685
\(999\) 0 0
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 6003.2.a.m.1.10 11
3.2 odd 2 2001.2.a.l.1.2 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.l.1.2 11 3.2 odd 2
6003.2.a.m.1.10 11 1.1 even 1 trivial