Properties

Label 6003.2.a.m.1.1
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 1.1
Root \(2.72479\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.72479 q^{2} +5.42450 q^{4} +1.02492 q^{5} +4.32726 q^{7} -9.33107 q^{8} +O(q^{10})\) \(q-2.72479 q^{2} +5.42450 q^{4} +1.02492 q^{5} +4.32726 q^{7} -9.33107 q^{8} -2.79269 q^{10} -2.38338 q^{11} +0.754393 q^{13} -11.7909 q^{14} +14.5762 q^{16} +1.03913 q^{17} -8.17626 q^{19} +5.55967 q^{20} +6.49422 q^{22} -1.00000 q^{23} -3.94954 q^{25} -2.05557 q^{26} +23.4733 q^{28} +1.00000 q^{29} -5.92951 q^{31} -21.0551 q^{32} -2.83143 q^{34} +4.43509 q^{35} +3.05932 q^{37} +22.2786 q^{38} -9.56358 q^{40} -1.01080 q^{41} +8.18673 q^{43} -12.9287 q^{44} +2.72479 q^{46} -8.08697 q^{47} +11.7252 q^{49} +10.7617 q^{50} +4.09221 q^{52} +0.478182 q^{53} -2.44277 q^{55} -40.3780 q^{56} -2.72479 q^{58} +8.09543 q^{59} -14.8897 q^{61} +16.1567 q^{62} +28.2183 q^{64} +0.773191 q^{65} -0.830191 q^{67} +5.63679 q^{68} -12.0847 q^{70} +1.79880 q^{71} +13.2708 q^{73} -8.33601 q^{74} -44.3521 q^{76} -10.3135 q^{77} +2.46332 q^{79} +14.9394 q^{80} +2.75423 q^{82} -4.46805 q^{83} +1.06503 q^{85} -22.3072 q^{86} +22.2395 q^{88} +5.57787 q^{89} +3.26446 q^{91} -5.42450 q^{92} +22.0353 q^{94} -8.37999 q^{95} -0.684597 q^{97} -31.9488 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 2q^{2} + 18q^{4} - 2q^{5} + 3q^{7} - 18q^{8} + O(q^{10}) \) \( 11q - 2q^{2} + 18q^{4} - 2q^{5} + 3q^{7} - 18q^{8} + 14q^{10} - 11q^{11} - 5q^{13} - 17q^{14} + 20q^{16} - 15q^{17} - 6q^{19} - 21q^{20} - 10q^{22} - 11q^{23} + 3q^{25} + 5q^{26} + 7q^{28} + 11q^{29} + 35q^{31} - 28q^{32} + 28q^{34} - 15q^{35} - 28q^{37} + 2q^{38} - q^{40} - 10q^{41} - 6q^{43} - 18q^{44} + 2q^{46} - 15q^{47} + 22q^{49} - 15q^{50} - 36q^{52} + 7q^{53} - 12q^{55} - 56q^{56} - 2q^{58} + 20q^{59} - 20q^{61} + 11q^{62} + 36q^{64} - 11q^{65} - 39q^{67} - 35q^{68} + 38q^{70} - 49q^{71} - 3q^{73} - 37q^{74} - 18q^{76} - 25q^{77} + 41q^{79} - 51q^{80} - 19q^{82} - 13q^{83} - 62q^{86} - 40q^{88} - 34q^{89} + 2q^{91} - 18q^{92} - 14q^{94} - 25q^{95} - 11q^{97} - 53q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.72479 −1.92672 −0.963360 0.268211i \(-0.913568\pi\)
−0.963360 + 0.268211i \(0.913568\pi\)
\(3\) 0 0
\(4\) 5.42450 2.71225
\(5\) 1.02492 0.458357 0.229179 0.973384i \(-0.426396\pi\)
0.229179 + 0.973384i \(0.426396\pi\)
\(6\) 0 0
\(7\) 4.32726 1.63555 0.817776 0.575537i \(-0.195207\pi\)
0.817776 + 0.575537i \(0.195207\pi\)
\(8\) −9.33107 −3.29903
\(9\) 0 0
\(10\) −2.79269 −0.883126
\(11\) −2.38338 −0.718616 −0.359308 0.933219i \(-0.616987\pi\)
−0.359308 + 0.933219i \(0.616987\pi\)
\(12\) 0 0
\(13\) 0.754393 0.209231 0.104616 0.994513i \(-0.466639\pi\)
0.104616 + 0.994513i \(0.466639\pi\)
\(14\) −11.7909 −3.15125
\(15\) 0 0
\(16\) 14.5762 3.64406
\(17\) 1.03913 0.252027 0.126014 0.992029i \(-0.459782\pi\)
0.126014 + 0.992029i \(0.459782\pi\)
\(18\) 0 0
\(19\) −8.17626 −1.87576 −0.937881 0.346957i \(-0.887215\pi\)
−0.937881 + 0.346957i \(0.887215\pi\)
\(20\) 5.55967 1.24318
\(21\) 0 0
\(22\) 6.49422 1.38457
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.94954 −0.789909
\(26\) −2.05557 −0.403130
\(27\) 0 0
\(28\) 23.4733 4.43603
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −5.92951 −1.06497 −0.532485 0.846439i \(-0.678742\pi\)
−0.532485 + 0.846439i \(0.678742\pi\)
\(32\) −21.0551 −3.72205
\(33\) 0 0
\(34\) −2.83143 −0.485586
\(35\) 4.43509 0.749667
\(36\) 0 0
\(37\) 3.05932 0.502948 0.251474 0.967864i \(-0.419085\pi\)
0.251474 + 0.967864i \(0.419085\pi\)
\(38\) 22.2786 3.61407
\(39\) 0 0
\(40\) −9.56358 −1.51213
\(41\) −1.01080 −0.157861 −0.0789304 0.996880i \(-0.525150\pi\)
−0.0789304 + 0.996880i \(0.525150\pi\)
\(42\) 0 0
\(43\) 8.18673 1.24846 0.624232 0.781239i \(-0.285412\pi\)
0.624232 + 0.781239i \(0.285412\pi\)
\(44\) −12.9287 −1.94907
\(45\) 0 0
\(46\) 2.72479 0.401749
\(47\) −8.08697 −1.17961 −0.589803 0.807547i \(-0.700795\pi\)
−0.589803 + 0.807547i \(0.700795\pi\)
\(48\) 0 0
\(49\) 11.7252 1.67503
\(50\) 10.7617 1.52193
\(51\) 0 0
\(52\) 4.09221 0.567487
\(53\) 0.478182 0.0656834 0.0328417 0.999461i \(-0.489544\pi\)
0.0328417 + 0.999461i \(0.489544\pi\)
\(54\) 0 0
\(55\) −2.44277 −0.329383
\(56\) −40.3780 −5.39573
\(57\) 0 0
\(58\) −2.72479 −0.357783
\(59\) 8.09543 1.05394 0.526968 0.849885i \(-0.323329\pi\)
0.526968 + 0.849885i \(0.323329\pi\)
\(60\) 0 0
\(61\) −14.8897 −1.90644 −0.953218 0.302285i \(-0.902251\pi\)
−0.953218 + 0.302285i \(0.902251\pi\)
\(62\) 16.1567 2.05190
\(63\) 0 0
\(64\) 28.2183 3.52729
\(65\) 0.773191 0.0959026
\(66\) 0 0
\(67\) −0.830191 −0.101424 −0.0507120 0.998713i \(-0.516149\pi\)
−0.0507120 + 0.998713i \(0.516149\pi\)
\(68\) 5.63679 0.683561
\(69\) 0 0
\(70\) −12.0847 −1.44440
\(71\) 1.79880 0.213478 0.106739 0.994287i \(-0.465959\pi\)
0.106739 + 0.994287i \(0.465959\pi\)
\(72\) 0 0
\(73\) 13.2708 1.55323 0.776617 0.629973i \(-0.216934\pi\)
0.776617 + 0.629973i \(0.216934\pi\)
\(74\) −8.33601 −0.969041
\(75\) 0 0
\(76\) −44.3521 −5.08754
\(77\) −10.3135 −1.17533
\(78\) 0 0
\(79\) 2.46332 0.277145 0.138572 0.990352i \(-0.455749\pi\)
0.138572 + 0.990352i \(0.455749\pi\)
\(80\) 14.9394 1.67028
\(81\) 0 0
\(82\) 2.75423 0.304153
\(83\) −4.46805 −0.490432 −0.245216 0.969468i \(-0.578859\pi\)
−0.245216 + 0.969468i \(0.578859\pi\)
\(84\) 0 0
\(85\) 1.06503 0.115518
\(86\) −22.3072 −2.40544
\(87\) 0 0
\(88\) 22.2395 2.37074
\(89\) 5.57787 0.591253 0.295626 0.955304i \(-0.404472\pi\)
0.295626 + 0.955304i \(0.404472\pi\)
\(90\) 0 0
\(91\) 3.26446 0.342208
\(92\) −5.42450 −0.565544
\(93\) 0 0
\(94\) 22.0353 2.27277
\(95\) −8.37999 −0.859769
\(96\) 0 0
\(97\) −0.684597 −0.0695103 −0.0347551 0.999396i \(-0.511065\pi\)
−0.0347551 + 0.999396i \(0.511065\pi\)
\(98\) −31.9488 −3.22731
\(99\) 0 0
\(100\) −21.4243 −2.14243
\(101\) −19.2544 −1.91589 −0.957943 0.286958i \(-0.907356\pi\)
−0.957943 + 0.286958i \(0.907356\pi\)
\(102\) 0 0
\(103\) −5.09888 −0.502408 −0.251204 0.967934i \(-0.580826\pi\)
−0.251204 + 0.967934i \(0.580826\pi\)
\(104\) −7.03929 −0.690260
\(105\) 0 0
\(106\) −1.30295 −0.126553
\(107\) 13.7603 1.33026 0.665131 0.746727i \(-0.268376\pi\)
0.665131 + 0.746727i \(0.268376\pi\)
\(108\) 0 0
\(109\) −12.0945 −1.15845 −0.579223 0.815169i \(-0.696644\pi\)
−0.579223 + 0.815169i \(0.696644\pi\)
\(110\) 6.65604 0.634629
\(111\) 0 0
\(112\) 63.0752 5.96005
\(113\) −3.66701 −0.344963 −0.172482 0.985013i \(-0.555179\pi\)
−0.172482 + 0.985013i \(0.555179\pi\)
\(114\) 0 0
\(115\) −1.02492 −0.0955741
\(116\) 5.42450 0.503653
\(117\) 0 0
\(118\) −22.0584 −2.03064
\(119\) 4.49661 0.412203
\(120\) 0 0
\(121\) −5.31950 −0.483591
\(122\) 40.5715 3.67317
\(123\) 0 0
\(124\) −32.1646 −2.88847
\(125\) −9.17255 −0.820418
\(126\) 0 0
\(127\) −9.23920 −0.819847 −0.409923 0.912120i \(-0.634445\pi\)
−0.409923 + 0.912120i \(0.634445\pi\)
\(128\) −34.7790 −3.07406
\(129\) 0 0
\(130\) −2.10679 −0.184777
\(131\) 6.81925 0.595800 0.297900 0.954597i \(-0.403714\pi\)
0.297900 + 0.954597i \(0.403714\pi\)
\(132\) 0 0
\(133\) −35.3808 −3.06791
\(134\) 2.26210 0.195416
\(135\) 0 0
\(136\) −9.69623 −0.831445
\(137\) −4.86061 −0.415270 −0.207635 0.978206i \(-0.566577\pi\)
−0.207635 + 0.978206i \(0.566577\pi\)
\(138\) 0 0
\(139\) −11.5906 −0.983105 −0.491553 0.870848i \(-0.663570\pi\)
−0.491553 + 0.870848i \(0.663570\pi\)
\(140\) 24.0582 2.03329
\(141\) 0 0
\(142\) −4.90136 −0.411313
\(143\) −1.79801 −0.150357
\(144\) 0 0
\(145\) 1.02492 0.0851148
\(146\) −36.1603 −2.99265
\(147\) 0 0
\(148\) 16.5953 1.36412
\(149\) 5.07301 0.415597 0.207799 0.978172i \(-0.433370\pi\)
0.207799 + 0.978172i \(0.433370\pi\)
\(150\) 0 0
\(151\) 9.42334 0.766861 0.383430 0.923570i \(-0.374743\pi\)
0.383430 + 0.923570i \(0.374743\pi\)
\(152\) 76.2932 6.18820
\(153\) 0 0
\(154\) 28.1022 2.26454
\(155\) −6.07726 −0.488137
\(156\) 0 0
\(157\) −17.9873 −1.43554 −0.717772 0.696278i \(-0.754838\pi\)
−0.717772 + 0.696278i \(0.754838\pi\)
\(158\) −6.71204 −0.533981
\(159\) 0 0
\(160\) −21.5798 −1.70603
\(161\) −4.32726 −0.341036
\(162\) 0 0
\(163\) −16.3947 −1.28413 −0.642066 0.766649i \(-0.721922\pi\)
−0.642066 + 0.766649i \(0.721922\pi\)
\(164\) −5.48310 −0.428158
\(165\) 0 0
\(166\) 12.1745 0.944925
\(167\) −22.7384 −1.75955 −0.879775 0.475390i \(-0.842307\pi\)
−0.879775 + 0.475390i \(0.842307\pi\)
\(168\) 0 0
\(169\) −12.4309 −0.956222
\(170\) −2.90198 −0.222572
\(171\) 0 0
\(172\) 44.4090 3.38615
\(173\) −21.4566 −1.63132 −0.815659 0.578533i \(-0.803625\pi\)
−0.815659 + 0.578533i \(0.803625\pi\)
\(174\) 0 0
\(175\) −17.0907 −1.29194
\(176\) −34.7407 −2.61868
\(177\) 0 0
\(178\) −15.1985 −1.13918
\(179\) 14.9868 1.12016 0.560082 0.828437i \(-0.310769\pi\)
0.560082 + 0.828437i \(0.310769\pi\)
\(180\) 0 0
\(181\) −17.0940 −1.27059 −0.635295 0.772269i \(-0.719122\pi\)
−0.635295 + 0.772269i \(0.719122\pi\)
\(182\) −8.89498 −0.659340
\(183\) 0 0
\(184\) 9.33107 0.687895
\(185\) 3.13555 0.230530
\(186\) 0 0
\(187\) −2.47665 −0.181111
\(188\) −43.8678 −3.19939
\(189\) 0 0
\(190\) 22.8338 1.65653
\(191\) 15.6430 1.13189 0.565944 0.824444i \(-0.308512\pi\)
0.565944 + 0.824444i \(0.308512\pi\)
\(192\) 0 0
\(193\) 21.1145 1.51985 0.759927 0.650009i \(-0.225235\pi\)
0.759927 + 0.650009i \(0.225235\pi\)
\(194\) 1.86538 0.133927
\(195\) 0 0
\(196\) 63.6034 4.54310
\(197\) −2.37520 −0.169226 −0.0846128 0.996414i \(-0.526965\pi\)
−0.0846128 + 0.996414i \(0.526965\pi\)
\(198\) 0 0
\(199\) −14.1714 −1.00459 −0.502294 0.864697i \(-0.667510\pi\)
−0.502294 + 0.864697i \(0.667510\pi\)
\(200\) 36.8535 2.60593
\(201\) 0 0
\(202\) 52.4643 3.69138
\(203\) 4.32726 0.303714
\(204\) 0 0
\(205\) −1.03599 −0.0723566
\(206\) 13.8934 0.968000
\(207\) 0 0
\(208\) 10.9962 0.762450
\(209\) 19.4871 1.34795
\(210\) 0 0
\(211\) 12.8837 0.886950 0.443475 0.896287i \(-0.353746\pi\)
0.443475 + 0.896287i \(0.353746\pi\)
\(212\) 2.59390 0.178150
\(213\) 0 0
\(214\) −37.4941 −2.56304
\(215\) 8.39073 0.572243
\(216\) 0 0
\(217\) −25.6585 −1.74181
\(218\) 32.9551 2.23200
\(219\) 0 0
\(220\) −13.2508 −0.893369
\(221\) 0.783916 0.0527319
\(222\) 0 0
\(223\) −3.16955 −0.212249 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(224\) −91.1109 −6.08761
\(225\) 0 0
\(226\) 9.99184 0.664647
\(227\) −23.4740 −1.55802 −0.779012 0.627009i \(-0.784279\pi\)
−0.779012 + 0.627009i \(0.784279\pi\)
\(228\) 0 0
\(229\) 12.4848 0.825019 0.412510 0.910953i \(-0.364652\pi\)
0.412510 + 0.910953i \(0.364652\pi\)
\(230\) 2.79269 0.184145
\(231\) 0 0
\(232\) −9.33107 −0.612615
\(233\) 9.55710 0.626107 0.313053 0.949736i \(-0.398648\pi\)
0.313053 + 0.949736i \(0.398648\pi\)
\(234\) 0 0
\(235\) −8.28849 −0.540681
\(236\) 43.9137 2.85854
\(237\) 0 0
\(238\) −12.2523 −0.794201
\(239\) −2.85259 −0.184519 −0.0922593 0.995735i \(-0.529409\pi\)
−0.0922593 + 0.995735i \(0.529409\pi\)
\(240\) 0 0
\(241\) 0.712445 0.0458926 0.0229463 0.999737i \(-0.492695\pi\)
0.0229463 + 0.999737i \(0.492695\pi\)
\(242\) 14.4946 0.931745
\(243\) 0 0
\(244\) −80.7694 −5.17073
\(245\) 12.0174 0.767762
\(246\) 0 0
\(247\) −6.16811 −0.392468
\(248\) 55.3286 3.51337
\(249\) 0 0
\(250\) 24.9933 1.58072
\(251\) 8.60983 0.543447 0.271724 0.962375i \(-0.412406\pi\)
0.271724 + 0.962375i \(0.412406\pi\)
\(252\) 0 0
\(253\) 2.38338 0.149842
\(254\) 25.1749 1.57962
\(255\) 0 0
\(256\) 38.3289 2.39556
\(257\) −21.7102 −1.35425 −0.677123 0.735870i \(-0.736773\pi\)
−0.677123 + 0.735870i \(0.736773\pi\)
\(258\) 0 0
\(259\) 13.2385 0.822598
\(260\) 4.19418 0.260112
\(261\) 0 0
\(262\) −18.5810 −1.14794
\(263\) 18.3399 1.13089 0.565443 0.824787i \(-0.308705\pi\)
0.565443 + 0.824787i \(0.308705\pi\)
\(264\) 0 0
\(265\) 0.490097 0.0301064
\(266\) 96.4054 5.91100
\(267\) 0 0
\(268\) −4.50337 −0.275087
\(269\) 12.5342 0.764222 0.382111 0.924116i \(-0.375197\pi\)
0.382111 + 0.924116i \(0.375197\pi\)
\(270\) 0 0
\(271\) 11.5839 0.703673 0.351837 0.936061i \(-0.385557\pi\)
0.351837 + 0.936061i \(0.385557\pi\)
\(272\) 15.1467 0.918401
\(273\) 0 0
\(274\) 13.2442 0.800110
\(275\) 9.41326 0.567641
\(276\) 0 0
\(277\) −19.8284 −1.19137 −0.595686 0.803217i \(-0.703120\pi\)
−0.595686 + 0.803217i \(0.703120\pi\)
\(278\) 31.5821 1.89417
\(279\) 0 0
\(280\) −41.3841 −2.47317
\(281\) 14.0464 0.837936 0.418968 0.908001i \(-0.362392\pi\)
0.418968 + 0.908001i \(0.362392\pi\)
\(282\) 0 0
\(283\) 3.86800 0.229929 0.114964 0.993370i \(-0.463325\pi\)
0.114964 + 0.993370i \(0.463325\pi\)
\(284\) 9.75760 0.579007
\(285\) 0 0
\(286\) 4.89919 0.289695
\(287\) −4.37400 −0.258189
\(288\) 0 0
\(289\) −15.9202 −0.936482
\(290\) −2.79269 −0.163992
\(291\) 0 0
\(292\) 71.9877 4.21276
\(293\) −24.7095 −1.44355 −0.721774 0.692129i \(-0.756673\pi\)
−0.721774 + 0.692129i \(0.756673\pi\)
\(294\) 0 0
\(295\) 8.29715 0.483079
\(296\) −28.5467 −1.65924
\(297\) 0 0
\(298\) −13.8229 −0.800740
\(299\) −0.754393 −0.0436277
\(300\) 0 0
\(301\) 35.4261 2.04193
\(302\) −25.6767 −1.47753
\(303\) 0 0
\(304\) −119.179 −6.83539
\(305\) −15.2608 −0.873829
\(306\) 0 0
\(307\) 28.6139 1.63308 0.816539 0.577290i \(-0.195890\pi\)
0.816539 + 0.577290i \(0.195890\pi\)
\(308\) −55.9457 −3.18780
\(309\) 0 0
\(310\) 16.5593 0.940504
\(311\) −9.41777 −0.534033 −0.267016 0.963692i \(-0.586038\pi\)
−0.267016 + 0.963692i \(0.586038\pi\)
\(312\) 0 0
\(313\) −2.66933 −0.150880 −0.0754398 0.997150i \(-0.524036\pi\)
−0.0754398 + 0.997150i \(0.524036\pi\)
\(314\) 49.0118 2.76589
\(315\) 0 0
\(316\) 13.3623 0.751687
\(317\) −16.2962 −0.915283 −0.457642 0.889137i \(-0.651306\pi\)
−0.457642 + 0.889137i \(0.651306\pi\)
\(318\) 0 0
\(319\) −2.38338 −0.133444
\(320\) 28.9215 1.61676
\(321\) 0 0
\(322\) 11.7909 0.657081
\(323\) −8.49623 −0.472743
\(324\) 0 0
\(325\) −2.97951 −0.165273
\(326\) 44.6722 2.47416
\(327\) 0 0
\(328\) 9.43186 0.520787
\(329\) −34.9945 −1.92931
\(330\) 0 0
\(331\) 12.7644 0.701597 0.350798 0.936451i \(-0.385910\pi\)
0.350798 + 0.936451i \(0.385910\pi\)
\(332\) −24.2369 −1.33018
\(333\) 0 0
\(334\) 61.9574 3.39016
\(335\) −0.850878 −0.0464884
\(336\) 0 0
\(337\) 23.1747 1.26240 0.631202 0.775618i \(-0.282562\pi\)
0.631202 + 0.775618i \(0.282562\pi\)
\(338\) 33.8716 1.84237
\(339\) 0 0
\(340\) 5.77725 0.313315
\(341\) 14.1323 0.765305
\(342\) 0 0
\(343\) 20.4472 1.10404
\(344\) −76.3910 −4.11872
\(345\) 0 0
\(346\) 58.4649 3.14309
\(347\) 7.87073 0.422523 0.211262 0.977430i \(-0.432243\pi\)
0.211262 + 0.977430i \(0.432243\pi\)
\(348\) 0 0
\(349\) −20.2614 −1.08457 −0.542284 0.840195i \(-0.682440\pi\)
−0.542284 + 0.840195i \(0.682440\pi\)
\(350\) 46.5687 2.48920
\(351\) 0 0
\(352\) 50.1823 2.67472
\(353\) 2.08029 0.110723 0.0553614 0.998466i \(-0.482369\pi\)
0.0553614 + 0.998466i \(0.482369\pi\)
\(354\) 0 0
\(355\) 1.84362 0.0978494
\(356\) 30.2572 1.60363
\(357\) 0 0
\(358\) −40.8359 −2.15824
\(359\) −24.0681 −1.27027 −0.635133 0.772403i \(-0.719055\pi\)
−0.635133 + 0.772403i \(0.719055\pi\)
\(360\) 0 0
\(361\) 47.8512 2.51848
\(362\) 46.5778 2.44807
\(363\) 0 0
\(364\) 17.7081 0.928155
\(365\) 13.6015 0.711936
\(366\) 0 0
\(367\) −7.21883 −0.376820 −0.188410 0.982090i \(-0.560333\pi\)
−0.188410 + 0.982090i \(0.560333\pi\)
\(368\) −14.5762 −0.759839
\(369\) 0 0
\(370\) −8.54372 −0.444167
\(371\) 2.06922 0.107429
\(372\) 0 0
\(373\) −20.7310 −1.07341 −0.536704 0.843770i \(-0.680331\pi\)
−0.536704 + 0.843770i \(0.680331\pi\)
\(374\) 6.74837 0.348950
\(375\) 0 0
\(376\) 75.4601 3.89156
\(377\) 0.754393 0.0388532
\(378\) 0 0
\(379\) 12.9895 0.667228 0.333614 0.942710i \(-0.391732\pi\)
0.333614 + 0.942710i \(0.391732\pi\)
\(380\) −45.4573 −2.33191
\(381\) 0 0
\(382\) −42.6240 −2.18083
\(383\) 35.6863 1.82349 0.911743 0.410762i \(-0.134737\pi\)
0.911743 + 0.410762i \(0.134737\pi\)
\(384\) 0 0
\(385\) −10.5705 −0.538723
\(386\) −57.5326 −2.92833
\(387\) 0 0
\(388\) −3.71360 −0.188529
\(389\) −1.85642 −0.0941243 −0.0470622 0.998892i \(-0.514986\pi\)
−0.0470622 + 0.998892i \(0.514986\pi\)
\(390\) 0 0
\(391\) −1.03913 −0.0525513
\(392\) −109.409 −5.52597
\(393\) 0 0
\(394\) 6.47192 0.326051
\(395\) 2.52470 0.127031
\(396\) 0 0
\(397\) −18.4801 −0.927489 −0.463745 0.885969i \(-0.653494\pi\)
−0.463745 + 0.885969i \(0.653494\pi\)
\(398\) 38.6143 1.93556
\(399\) 0 0
\(400\) −57.5695 −2.87847
\(401\) 24.3126 1.21411 0.607056 0.794659i \(-0.292350\pi\)
0.607056 + 0.794659i \(0.292350\pi\)
\(402\) 0 0
\(403\) −4.47318 −0.222825
\(404\) −104.446 −5.19637
\(405\) 0 0
\(406\) −11.7909 −0.585173
\(407\) −7.29151 −0.361427
\(408\) 0 0
\(409\) 30.3298 1.49971 0.749855 0.661602i \(-0.230123\pi\)
0.749855 + 0.661602i \(0.230123\pi\)
\(410\) 2.82286 0.139411
\(411\) 0 0
\(412\) −27.6589 −1.36266
\(413\) 35.0311 1.72377
\(414\) 0 0
\(415\) −4.57938 −0.224793
\(416\) −15.8838 −0.778769
\(417\) 0 0
\(418\) −53.0984 −2.59713
\(419\) −27.8837 −1.36221 −0.681105 0.732186i \(-0.738500\pi\)
−0.681105 + 0.732186i \(0.738500\pi\)
\(420\) 0 0
\(421\) 14.0024 0.682437 0.341219 0.939984i \(-0.389160\pi\)
0.341219 + 0.939984i \(0.389160\pi\)
\(422\) −35.1054 −1.70890
\(423\) 0 0
\(424\) −4.46195 −0.216691
\(425\) −4.10411 −0.199078
\(426\) 0 0
\(427\) −64.4318 −3.11807
\(428\) 74.6430 3.60801
\(429\) 0 0
\(430\) −22.8630 −1.10255
\(431\) −29.8983 −1.44015 −0.720075 0.693897i \(-0.755892\pi\)
−0.720075 + 0.693897i \(0.755892\pi\)
\(432\) 0 0
\(433\) −37.3902 −1.79686 −0.898428 0.439120i \(-0.855290\pi\)
−0.898428 + 0.439120i \(0.855290\pi\)
\(434\) 69.9142 3.35599
\(435\) 0 0
\(436\) −65.6068 −3.14200
\(437\) 8.17626 0.391123
\(438\) 0 0
\(439\) −27.9734 −1.33510 −0.667549 0.744565i \(-0.732657\pi\)
−0.667549 + 0.744565i \(0.732657\pi\)
\(440\) 22.7936 1.08664
\(441\) 0 0
\(442\) −2.13601 −0.101600
\(443\) −12.5665 −0.597051 −0.298525 0.954402i \(-0.596495\pi\)
−0.298525 + 0.954402i \(0.596495\pi\)
\(444\) 0 0
\(445\) 5.71686 0.271005
\(446\) 8.63638 0.408944
\(447\) 0 0
\(448\) 122.108 5.76907
\(449\) −5.54208 −0.261547 −0.130774 0.991412i \(-0.541746\pi\)
−0.130774 + 0.991412i \(0.541746\pi\)
\(450\) 0 0
\(451\) 2.40912 0.113441
\(452\) −19.8917 −0.935627
\(453\) 0 0
\(454\) 63.9618 3.00188
\(455\) 3.34580 0.156854
\(456\) 0 0
\(457\) 32.2110 1.50677 0.753383 0.657582i \(-0.228421\pi\)
0.753383 + 0.657582i \(0.228421\pi\)
\(458\) −34.0185 −1.58958
\(459\) 0 0
\(460\) −5.55967 −0.259221
\(461\) 22.9127 1.06715 0.533576 0.845752i \(-0.320848\pi\)
0.533576 + 0.845752i \(0.320848\pi\)
\(462\) 0 0
\(463\) −1.70309 −0.0791491 −0.0395746 0.999217i \(-0.512600\pi\)
−0.0395746 + 0.999217i \(0.512600\pi\)
\(464\) 14.5762 0.676685
\(465\) 0 0
\(466\) −26.0411 −1.20633
\(467\) −28.0758 −1.29919 −0.649596 0.760279i \(-0.725062\pi\)
−0.649596 + 0.760279i \(0.725062\pi\)
\(468\) 0 0
\(469\) −3.59245 −0.165884
\(470\) 22.5844 1.04174
\(471\) 0 0
\(472\) −75.5390 −3.47697
\(473\) −19.5121 −0.897167
\(474\) 0 0
\(475\) 32.2925 1.48168
\(476\) 24.3919 1.11800
\(477\) 0 0
\(478\) 7.77272 0.355516
\(479\) 25.5753 1.16856 0.584282 0.811551i \(-0.301376\pi\)
0.584282 + 0.811551i \(0.301376\pi\)
\(480\) 0 0
\(481\) 2.30793 0.105232
\(482\) −1.94127 −0.0884223
\(483\) 0 0
\(484\) −28.8557 −1.31162
\(485\) −0.701655 −0.0318605
\(486\) 0 0
\(487\) 34.5635 1.56622 0.783112 0.621881i \(-0.213631\pi\)
0.783112 + 0.621881i \(0.213631\pi\)
\(488\) 138.937 6.28939
\(489\) 0 0
\(490\) −32.7449 −1.47926
\(491\) −14.9939 −0.676664 −0.338332 0.941027i \(-0.609863\pi\)
−0.338332 + 0.941027i \(0.609863\pi\)
\(492\) 0 0
\(493\) 1.03913 0.0468003
\(494\) 16.8068 0.756175
\(495\) 0 0
\(496\) −86.4299 −3.88082
\(497\) 7.78388 0.349155
\(498\) 0 0
\(499\) 1.58115 0.0707819 0.0353909 0.999374i \(-0.488732\pi\)
0.0353909 + 0.999374i \(0.488732\pi\)
\(500\) −49.7565 −2.22518
\(501\) 0 0
\(502\) −23.4600 −1.04707
\(503\) −38.2023 −1.70336 −0.851679 0.524064i \(-0.824415\pi\)
−0.851679 + 0.524064i \(0.824415\pi\)
\(504\) 0 0
\(505\) −19.7342 −0.878161
\(506\) −6.49422 −0.288703
\(507\) 0 0
\(508\) −50.1181 −2.22363
\(509\) −0.866481 −0.0384061 −0.0192030 0.999816i \(-0.506113\pi\)
−0.0192030 + 0.999816i \(0.506113\pi\)
\(510\) 0 0
\(511\) 57.4264 2.54039
\(512\) −34.8804 −1.54151
\(513\) 0 0
\(514\) 59.1558 2.60925
\(515\) −5.22594 −0.230282
\(516\) 0 0
\(517\) 19.2743 0.847684
\(518\) −36.0721 −1.58492
\(519\) 0 0
\(520\) −7.21470 −0.316386
\(521\) −3.68724 −0.161541 −0.0807704 0.996733i \(-0.525738\pi\)
−0.0807704 + 0.996733i \(0.525738\pi\)
\(522\) 0 0
\(523\) −41.5135 −1.81526 −0.907628 0.419775i \(-0.862109\pi\)
−0.907628 + 0.419775i \(0.862109\pi\)
\(524\) 36.9910 1.61596
\(525\) 0 0
\(526\) −49.9724 −2.17890
\(527\) −6.16155 −0.268401
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −1.33541 −0.0580067
\(531\) 0 0
\(532\) −191.923 −8.32093
\(533\) −0.762542 −0.0330294
\(534\) 0 0
\(535\) 14.1032 0.609735
\(536\) 7.74657 0.334601
\(537\) 0 0
\(538\) −34.1530 −1.47244
\(539\) −27.9456 −1.20370
\(540\) 0 0
\(541\) −25.7510 −1.10712 −0.553561 0.832809i \(-0.686731\pi\)
−0.553561 + 0.832809i \(0.686731\pi\)
\(542\) −31.5638 −1.35578
\(543\) 0 0
\(544\) −21.8791 −0.938058
\(545\) −12.3959 −0.530982
\(546\) 0 0
\(547\) 28.4992 1.21854 0.609268 0.792964i \(-0.291463\pi\)
0.609268 + 0.792964i \(0.291463\pi\)
\(548\) −26.3664 −1.12632
\(549\) 0 0
\(550\) −25.6492 −1.09369
\(551\) −8.17626 −0.348320
\(552\) 0 0
\(553\) 10.6594 0.453285
\(554\) 54.0283 2.29544
\(555\) 0 0
\(556\) −62.8735 −2.66643
\(557\) 29.0231 1.22975 0.614873 0.788626i \(-0.289207\pi\)
0.614873 + 0.788626i \(0.289207\pi\)
\(558\) 0 0
\(559\) 6.17602 0.261218
\(560\) 64.6469 2.73183
\(561\) 0 0
\(562\) −38.2735 −1.61447
\(563\) 30.2427 1.27458 0.637288 0.770626i \(-0.280056\pi\)
0.637288 + 0.770626i \(0.280056\pi\)
\(564\) 0 0
\(565\) −3.75838 −0.158116
\(566\) −10.5395 −0.443009
\(567\) 0 0
\(568\) −16.7847 −0.704272
\(569\) 34.0959 1.42937 0.714687 0.699444i \(-0.246569\pi\)
0.714687 + 0.699444i \(0.246569\pi\)
\(570\) 0 0
\(571\) −7.96785 −0.333444 −0.166722 0.986004i \(-0.553318\pi\)
−0.166722 + 0.986004i \(0.553318\pi\)
\(572\) −9.75329 −0.407805
\(573\) 0 0
\(574\) 11.9183 0.497459
\(575\) 3.94954 0.164707
\(576\) 0 0
\(577\) −35.2407 −1.46709 −0.733545 0.679640i \(-0.762136\pi\)
−0.733545 + 0.679640i \(0.762136\pi\)
\(578\) 43.3793 1.80434
\(579\) 0 0
\(580\) 5.55967 0.230853
\(581\) −19.3344 −0.802127
\(582\) 0 0
\(583\) −1.13969 −0.0472011
\(584\) −123.831 −5.12416
\(585\) 0 0
\(586\) 67.3284 2.78131
\(587\) 6.85699 0.283018 0.141509 0.989937i \(-0.454805\pi\)
0.141509 + 0.989937i \(0.454805\pi\)
\(588\) 0 0
\(589\) 48.4812 1.99763
\(590\) −22.6080 −0.930758
\(591\) 0 0
\(592\) 44.5933 1.83277
\(593\) −36.7362 −1.50858 −0.754288 0.656543i \(-0.772018\pi\)
−0.754288 + 0.656543i \(0.772018\pi\)
\(594\) 0 0
\(595\) 4.60865 0.188936
\(596\) 27.5186 1.12720
\(597\) 0 0
\(598\) 2.05557 0.0840584
\(599\) −39.6697 −1.62086 −0.810430 0.585836i \(-0.800766\pi\)
−0.810430 + 0.585836i \(0.800766\pi\)
\(600\) 0 0
\(601\) −13.4001 −0.546601 −0.273300 0.961929i \(-0.588115\pi\)
−0.273300 + 0.961929i \(0.588115\pi\)
\(602\) −96.5289 −3.93423
\(603\) 0 0
\(604\) 51.1169 2.07992
\(605\) −5.45205 −0.221658
\(606\) 0 0
\(607\) 10.1830 0.413313 0.206657 0.978414i \(-0.433742\pi\)
0.206657 + 0.978414i \(0.433742\pi\)
\(608\) 172.152 6.98168
\(609\) 0 0
\(610\) 41.5824 1.68362
\(611\) −6.10076 −0.246810
\(612\) 0 0
\(613\) −9.64274 −0.389467 −0.194733 0.980856i \(-0.562384\pi\)
−0.194733 + 0.980856i \(0.562384\pi\)
\(614\) −77.9669 −3.14649
\(615\) 0 0
\(616\) 96.2361 3.87746
\(617\) 1.00133 0.0403122 0.0201561 0.999797i \(-0.493584\pi\)
0.0201561 + 0.999797i \(0.493584\pi\)
\(618\) 0 0
\(619\) 29.4218 1.18256 0.591281 0.806466i \(-0.298622\pi\)
0.591281 + 0.806466i \(0.298622\pi\)
\(620\) −32.9661 −1.32395
\(621\) 0 0
\(622\) 25.6615 1.02893
\(623\) 24.1369 0.967025
\(624\) 0 0
\(625\) 10.3466 0.413864
\(626\) 7.27338 0.290703
\(627\) 0 0
\(628\) −97.5723 −3.89356
\(629\) 3.17904 0.126757
\(630\) 0 0
\(631\) −46.2853 −1.84259 −0.921294 0.388866i \(-0.872867\pi\)
−0.921294 + 0.388866i \(0.872867\pi\)
\(632\) −22.9854 −0.914310
\(633\) 0 0
\(634\) 44.4037 1.76349
\(635\) −9.46943 −0.375783
\(636\) 0 0
\(637\) 8.84541 0.350468
\(638\) 6.49422 0.257109
\(639\) 0 0
\(640\) −35.6456 −1.40902
\(641\) −29.5602 −1.16756 −0.583780 0.811912i \(-0.698427\pi\)
−0.583780 + 0.811912i \(0.698427\pi\)
\(642\) 0 0
\(643\) 22.8121 0.899621 0.449810 0.893124i \(-0.351492\pi\)
0.449810 + 0.893124i \(0.351492\pi\)
\(644\) −23.4733 −0.924976
\(645\) 0 0
\(646\) 23.1505 0.910843
\(647\) −27.7127 −1.08950 −0.544749 0.838599i \(-0.683375\pi\)
−0.544749 + 0.838599i \(0.683375\pi\)
\(648\) 0 0
\(649\) −19.2945 −0.757375
\(650\) 8.11855 0.318436
\(651\) 0 0
\(652\) −88.9331 −3.48289
\(653\) 18.2397 0.713774 0.356887 0.934148i \(-0.383838\pi\)
0.356887 + 0.934148i \(0.383838\pi\)
\(654\) 0 0
\(655\) 6.98917 0.273089
\(656\) −14.7337 −0.575254
\(657\) 0 0
\(658\) 95.3527 3.71724
\(659\) −18.0505 −0.703149 −0.351574 0.936160i \(-0.614354\pi\)
−0.351574 + 0.936160i \(0.614354\pi\)
\(660\) 0 0
\(661\) −9.33849 −0.363225 −0.181613 0.983370i \(-0.558132\pi\)
−0.181613 + 0.983370i \(0.558132\pi\)
\(662\) −34.7805 −1.35178
\(663\) 0 0
\(664\) 41.6917 1.61795
\(665\) −36.2624 −1.40620
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) −123.345 −4.77234
\(669\) 0 0
\(670\) 2.31847 0.0895702
\(671\) 35.4879 1.36999
\(672\) 0 0
\(673\) 12.8952 0.497074 0.248537 0.968622i \(-0.420050\pi\)
0.248537 + 0.968622i \(0.420050\pi\)
\(674\) −63.1462 −2.43230
\(675\) 0 0
\(676\) −67.4314 −2.59352
\(677\) −45.6983 −1.75633 −0.878165 0.478358i \(-0.841232\pi\)
−0.878165 + 0.478358i \(0.841232\pi\)
\(678\) 0 0
\(679\) −2.96243 −0.113688
\(680\) −9.93784 −0.381099
\(681\) 0 0
\(682\) −38.5075 −1.47453
\(683\) 21.1386 0.808847 0.404423 0.914572i \(-0.367472\pi\)
0.404423 + 0.914572i \(0.367472\pi\)
\(684\) 0 0
\(685\) −4.98173 −0.190342
\(686\) −55.7144 −2.12719
\(687\) 0 0
\(688\) 119.332 4.54948
\(689\) 0.360737 0.0137430
\(690\) 0 0
\(691\) 10.9159 0.415261 0.207630 0.978207i \(-0.433425\pi\)
0.207630 + 0.978207i \(0.433425\pi\)
\(692\) −116.392 −4.42455
\(693\) 0 0
\(694\) −21.4461 −0.814084
\(695\) −11.8795 −0.450614
\(696\) 0 0
\(697\) −1.05036 −0.0397852
\(698\) 55.2081 2.08966
\(699\) 0 0
\(700\) −92.7086 −3.50406
\(701\) 12.4276 0.469384 0.234692 0.972070i \(-0.424592\pi\)
0.234692 + 0.972070i \(0.424592\pi\)
\(702\) 0 0
\(703\) −25.0138 −0.943411
\(704\) −67.2550 −2.53477
\(705\) 0 0
\(706\) −5.66837 −0.213332
\(707\) −83.3189 −3.13353
\(708\) 0 0
\(709\) −45.4539 −1.70706 −0.853529 0.521046i \(-0.825542\pi\)
−0.853529 + 0.521046i \(0.825542\pi\)
\(710\) −5.02350 −0.188528
\(711\) 0 0
\(712\) −52.0475 −1.95056
\(713\) 5.92951 0.222062
\(714\) 0 0
\(715\) −1.84281 −0.0689171
\(716\) 81.2959 3.03817
\(717\) 0 0
\(718\) 65.5806 2.44745
\(719\) 30.5357 1.13879 0.569394 0.822065i \(-0.307178\pi\)
0.569394 + 0.822065i \(0.307178\pi\)
\(720\) 0 0
\(721\) −22.0642 −0.821714
\(722\) −130.385 −4.85241
\(723\) 0 0
\(724\) −92.7267 −3.44616
\(725\) −3.94954 −0.146682
\(726\) 0 0
\(727\) 13.9153 0.516091 0.258045 0.966133i \(-0.416922\pi\)
0.258045 + 0.966133i \(0.416922\pi\)
\(728\) −30.4609 −1.12896
\(729\) 0 0
\(730\) −37.0613 −1.37170
\(731\) 8.50711 0.314647
\(732\) 0 0
\(733\) 7.18360 0.265332 0.132666 0.991161i \(-0.457646\pi\)
0.132666 + 0.991161i \(0.457646\pi\)
\(734\) 19.6698 0.726027
\(735\) 0 0
\(736\) 21.0551 0.776101
\(737\) 1.97866 0.0728849
\(738\) 0 0
\(739\) 3.44007 0.126545 0.0632725 0.997996i \(-0.479846\pi\)
0.0632725 + 0.997996i \(0.479846\pi\)
\(740\) 17.0088 0.625256
\(741\) 0 0
\(742\) −5.63820 −0.206985
\(743\) 6.89471 0.252942 0.126471 0.991970i \(-0.459635\pi\)
0.126471 + 0.991970i \(0.459635\pi\)
\(744\) 0 0
\(745\) 5.19942 0.190492
\(746\) 56.4876 2.06816
\(747\) 0 0
\(748\) −13.4346 −0.491218
\(749\) 59.5446 2.17571
\(750\) 0 0
\(751\) 44.3395 1.61797 0.808985 0.587829i \(-0.200017\pi\)
0.808985 + 0.587829i \(0.200017\pi\)
\(752\) −117.878 −4.29855
\(753\) 0 0
\(754\) −2.05557 −0.0748593
\(755\) 9.65815 0.351496
\(756\) 0 0
\(757\) −30.4467 −1.10660 −0.553302 0.832981i \(-0.686633\pi\)
−0.553302 + 0.832981i \(0.686633\pi\)
\(758\) −35.3938 −1.28556
\(759\) 0 0
\(760\) 78.1943 2.83640
\(761\) −34.4231 −1.24784 −0.623919 0.781489i \(-0.714460\pi\)
−0.623919 + 0.781489i \(0.714460\pi\)
\(762\) 0 0
\(763\) −52.3362 −1.89470
\(764\) 84.8555 3.06997
\(765\) 0 0
\(766\) −97.2378 −3.51335
\(767\) 6.10714 0.220516
\(768\) 0 0
\(769\) −34.2773 −1.23607 −0.618035 0.786151i \(-0.712071\pi\)
−0.618035 + 0.786151i \(0.712071\pi\)
\(770\) 28.8024 1.03797
\(771\) 0 0
\(772\) 114.536 4.12223
\(773\) 10.4729 0.376682 0.188341 0.982104i \(-0.439689\pi\)
0.188341 + 0.982104i \(0.439689\pi\)
\(774\) 0 0
\(775\) 23.4188 0.841229
\(776\) 6.38802 0.229316
\(777\) 0 0
\(778\) 5.05837 0.181351
\(779\) 8.26457 0.296109
\(780\) 0 0
\(781\) −4.28723 −0.153409
\(782\) 2.83143 0.101252
\(783\) 0 0
\(784\) 170.909 6.10390
\(785\) −18.4355 −0.657993
\(786\) 0 0
\(787\) 43.2021 1.53999 0.769995 0.638050i \(-0.220259\pi\)
0.769995 + 0.638050i \(0.220259\pi\)
\(788\) −12.8843 −0.458983
\(789\) 0 0
\(790\) −6.87929 −0.244754
\(791\) −15.8681 −0.564205
\(792\) 0 0
\(793\) −11.2327 −0.398885
\(794\) 50.3544 1.78701
\(795\) 0 0
\(796\) −76.8731 −2.72469
\(797\) 22.4875 0.796549 0.398274 0.917266i \(-0.369609\pi\)
0.398274 + 0.917266i \(0.369609\pi\)
\(798\) 0 0
\(799\) −8.40345 −0.297293
\(800\) 83.1580 2.94008
\(801\) 0 0
\(802\) −66.2468 −2.33926
\(803\) −31.6294 −1.11618
\(804\) 0 0
\(805\) −4.43509 −0.156316
\(806\) 12.1885 0.429321
\(807\) 0 0
\(808\) 179.664 6.32057
\(809\) −14.7365 −0.518108 −0.259054 0.965863i \(-0.583411\pi\)
−0.259054 + 0.965863i \(0.583411\pi\)
\(810\) 0 0
\(811\) 53.4867 1.87817 0.939086 0.343682i \(-0.111674\pi\)
0.939086 + 0.343682i \(0.111674\pi\)
\(812\) 23.4733 0.823750
\(813\) 0 0
\(814\) 19.8679 0.696368
\(815\) −16.8032 −0.588591
\(816\) 0 0
\(817\) −66.9368 −2.34182
\(818\) −82.6423 −2.88952
\(819\) 0 0
\(820\) −5.61973 −0.196249
\(821\) −21.2503 −0.741640 −0.370820 0.928705i \(-0.620923\pi\)
−0.370820 + 0.928705i \(0.620923\pi\)
\(822\) 0 0
\(823\) −11.6892 −0.407459 −0.203730 0.979027i \(-0.565306\pi\)
−0.203730 + 0.979027i \(0.565306\pi\)
\(824\) 47.5780 1.65746
\(825\) 0 0
\(826\) −95.4524 −3.32121
\(827\) 36.0442 1.25338 0.626690 0.779269i \(-0.284409\pi\)
0.626690 + 0.779269i \(0.284409\pi\)
\(828\) 0 0
\(829\) 11.0806 0.384846 0.192423 0.981312i \(-0.438366\pi\)
0.192423 + 0.981312i \(0.438366\pi\)
\(830\) 12.4779 0.433113
\(831\) 0 0
\(832\) 21.2877 0.738019
\(833\) 12.1841 0.422153
\(834\) 0 0
\(835\) −23.3050 −0.806503
\(836\) 105.708 3.65599
\(837\) 0 0
\(838\) 75.9775 2.62460
\(839\) −40.9744 −1.41459 −0.707296 0.706917i \(-0.750085\pi\)
−0.707296 + 0.706917i \(0.750085\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −38.1538 −1.31487
\(843\) 0 0
\(844\) 69.8876 2.40563
\(845\) −12.7406 −0.438292
\(846\) 0 0
\(847\) −23.0189 −0.790938
\(848\) 6.97009 0.239354
\(849\) 0 0
\(850\) 11.1828 0.383568
\(851\) −3.05932 −0.104872
\(852\) 0 0
\(853\) 20.2611 0.693726 0.346863 0.937916i \(-0.387247\pi\)
0.346863 + 0.937916i \(0.387247\pi\)
\(854\) 175.563 6.00766
\(855\) 0 0
\(856\) −128.399 −4.38858
\(857\) −12.5526 −0.428789 −0.214395 0.976747i \(-0.568778\pi\)
−0.214395 + 0.976747i \(0.568778\pi\)
\(858\) 0 0
\(859\) −49.7481 −1.69738 −0.848692 0.528887i \(-0.822609\pi\)
−0.848692 + 0.528887i \(0.822609\pi\)
\(860\) 45.5155 1.55207
\(861\) 0 0
\(862\) 81.4666 2.77476
\(863\) −29.3777 −1.00003 −0.500015 0.866017i \(-0.666672\pi\)
−0.500015 + 0.866017i \(0.666672\pi\)
\(864\) 0 0
\(865\) −21.9913 −0.747727
\(866\) 101.881 3.46204
\(867\) 0 0
\(868\) −139.185 −4.72424
\(869\) −5.87102 −0.199161
\(870\) 0 0
\(871\) −0.626290 −0.0212210
\(872\) 112.855 3.82175
\(873\) 0 0
\(874\) −22.2786 −0.753585
\(875\) −39.6920 −1.34184
\(876\) 0 0
\(877\) 47.3882 1.60019 0.800093 0.599876i \(-0.204783\pi\)
0.800093 + 0.599876i \(0.204783\pi\)
\(878\) 76.2218 2.57236
\(879\) 0 0
\(880\) −35.6064 −1.20029
\(881\) 46.1479 1.55476 0.777382 0.629029i \(-0.216547\pi\)
0.777382 + 0.629029i \(0.216547\pi\)
\(882\) 0 0
\(883\) −17.8622 −0.601111 −0.300556 0.953764i \(-0.597172\pi\)
−0.300556 + 0.953764i \(0.597172\pi\)
\(884\) 4.25235 0.143022
\(885\) 0 0
\(886\) 34.2410 1.15035
\(887\) −34.0136 −1.14207 −0.571033 0.820927i \(-0.693457\pi\)
−0.571033 + 0.820927i \(0.693457\pi\)
\(888\) 0 0
\(889\) −39.9805 −1.34090
\(890\) −15.5773 −0.522151
\(891\) 0 0
\(892\) −17.1933 −0.575673
\(893\) 66.1212 2.21266
\(894\) 0 0
\(895\) 15.3602 0.513436
\(896\) −150.498 −5.02778
\(897\) 0 0
\(898\) 15.1010 0.503928
\(899\) −5.92951 −0.197760
\(900\) 0 0
\(901\) 0.496895 0.0165540
\(902\) −6.56437 −0.218570
\(903\) 0 0
\(904\) 34.2171 1.13804
\(905\) −17.5200 −0.582385
\(906\) 0 0
\(907\) 1.99413 0.0662141 0.0331070 0.999452i \(-0.489460\pi\)
0.0331070 + 0.999452i \(0.489460\pi\)
\(908\) −127.335 −4.22575
\(909\) 0 0
\(910\) −9.11662 −0.302213
\(911\) −3.41787 −0.113239 −0.0566196 0.998396i \(-0.518032\pi\)
−0.0566196 + 0.998396i \(0.518032\pi\)
\(912\) 0 0
\(913\) 10.6491 0.352432
\(914\) −87.7683 −2.90312
\(915\) 0 0
\(916\) 67.7239 2.23766
\(917\) 29.5087 0.974462
\(918\) 0 0
\(919\) 5.56763 0.183659 0.0918296 0.995775i \(-0.470729\pi\)
0.0918296 + 0.995775i \(0.470729\pi\)
\(920\) 9.56358 0.315302
\(921\) 0 0
\(922\) −62.4325 −2.05610
\(923\) 1.35700 0.0446663
\(924\) 0 0
\(925\) −12.0829 −0.397283
\(926\) 4.64056 0.152498
\(927\) 0 0
\(928\) −21.0551 −0.691167
\(929\) 32.3923 1.06276 0.531379 0.847134i \(-0.321674\pi\)
0.531379 + 0.847134i \(0.321674\pi\)
\(930\) 0 0
\(931\) −95.8683 −3.14196
\(932\) 51.8425 1.69816
\(933\) 0 0
\(934\) 76.5007 2.50318
\(935\) −2.53836 −0.0830134
\(936\) 0 0
\(937\) −1.53335 −0.0500925 −0.0250462 0.999686i \(-0.507973\pi\)
−0.0250462 + 0.999686i \(0.507973\pi\)
\(938\) 9.78870 0.319612
\(939\) 0 0
\(940\) −44.9609 −1.46646
\(941\) 4.27935 0.139503 0.0697514 0.997564i \(-0.477779\pi\)
0.0697514 + 0.997564i \(0.477779\pi\)
\(942\) 0 0
\(943\) 1.01080 0.0329162
\(944\) 118.001 3.84060
\(945\) 0 0
\(946\) 53.1664 1.72859
\(947\) −55.7160 −1.81053 −0.905263 0.424851i \(-0.860327\pi\)
−0.905263 + 0.424851i \(0.860327\pi\)
\(948\) 0 0
\(949\) 10.0114 0.324985
\(950\) −87.9903 −2.85478
\(951\) 0 0
\(952\) −41.9581 −1.35987
\(953\) −60.9052 −1.97291 −0.986457 0.164019i \(-0.947554\pi\)
−0.986457 + 0.164019i \(0.947554\pi\)
\(954\) 0 0
\(955\) 16.0328 0.518809
\(956\) −15.4739 −0.500461
\(957\) 0 0
\(958\) −69.6873 −2.25150
\(959\) −21.0332 −0.679196
\(960\) 0 0
\(961\) 4.15904 0.134163
\(962\) −6.28863 −0.202753
\(963\) 0 0
\(964\) 3.86466 0.124472
\(965\) 21.6406 0.696636
\(966\) 0 0
\(967\) 27.2667 0.876839 0.438420 0.898770i \(-0.355538\pi\)
0.438420 + 0.898770i \(0.355538\pi\)
\(968\) 49.6366 1.59538
\(969\) 0 0
\(970\) 1.91187 0.0613863
\(971\) −44.4273 −1.42574 −0.712870 0.701297i \(-0.752605\pi\)
−0.712870 + 0.701297i \(0.752605\pi\)
\(972\) 0 0
\(973\) −50.1558 −1.60792
\(974\) −94.1785 −3.01767
\(975\) 0 0
\(976\) −217.036 −6.94716
\(977\) −0.396649 −0.0126899 −0.00634496 0.999980i \(-0.502020\pi\)
−0.00634496 + 0.999980i \(0.502020\pi\)
\(978\) 0 0
\(979\) −13.2942 −0.424884
\(980\) 65.1883 2.08236
\(981\) 0 0
\(982\) 40.8552 1.30374
\(983\) 2.59076 0.0826323 0.0413161 0.999146i \(-0.486845\pi\)
0.0413161 + 0.999146i \(0.486845\pi\)
\(984\) 0 0
\(985\) −2.43438 −0.0775658
\(986\) −2.83143 −0.0901710
\(987\) 0 0
\(988\) −33.4590 −1.06447
\(989\) −8.18673 −0.260323
\(990\) 0 0
\(991\) 55.7666 1.77148 0.885742 0.464178i \(-0.153650\pi\)
0.885742 + 0.464178i \(0.153650\pi\)
\(992\) 124.846 3.96388
\(993\) 0 0
\(994\) −21.2095 −0.672724
\(995\) −14.5246 −0.460460
\(996\) 0 0
\(997\) −4.99638 −0.158237 −0.0791184 0.996865i \(-0.525211\pi\)
−0.0791184 + 0.996865i \(0.525211\pi\)
\(998\) −4.30830 −0.136377
\(999\) 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.1.1 11
3.2 odd 2 2001.2.a.l.1.11 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.l.1.11 11 3.2 odd 2
6003.2.a.m.1.1 11 1.1 even 1 trivial