Properties

Label 1521.2.a.n.1.1
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.69202 q^{2} +5.24698 q^{4} +1.04892 q^{5} +0.554958 q^{7} -8.74094 q^{8} +O(q^{10})\) \(q-2.69202 q^{2} +5.24698 q^{4} +1.04892 q^{5} +0.554958 q^{7} -8.74094 q^{8} -2.82371 q^{10} -2.91185 q^{11} -1.49396 q^{14} +13.0368 q^{16} +4.85086 q^{17} -0.753020 q^{19} +5.50365 q^{20} +7.83877 q^{22} -5.76271 q^{23} -3.89977 q^{25} +2.91185 q^{28} +1.91185 q^{29} -9.51573 q^{31} -17.6136 q^{32} -13.0586 q^{34} +0.582105 q^{35} +5.75302 q^{37} +2.02715 q^{38} -9.16852 q^{40} -4.91185 q^{41} -11.0978 q^{43} -15.2784 q^{44} +15.5133 q^{46} -0.753020 q^{47} -6.69202 q^{49} +10.4983 q^{50} +7.58211 q^{53} -3.05429 q^{55} -4.85086 q^{56} -5.14675 q^{58} -4.09783 q^{59} -3.42327 q^{61} +25.6165 q^{62} +21.3424 q^{64} -1.87263 q^{67} +25.4523 q^{68} -1.56704 q^{70} +10.5036 q^{71} +10.4765 q^{73} -15.4873 q^{74} -3.95108 q^{76} -1.61596 q^{77} +1.33513 q^{79} +13.6746 q^{80} +13.2228 q^{82} -2.64310 q^{83} +5.08815 q^{85} +29.8756 q^{86} +25.4523 q^{88} -9.92692 q^{89} -30.2368 q^{92} +2.02715 q^{94} -0.789856 q^{95} +17.0737 q^{97} +18.0151 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8} + O(q^{10}) \) \( 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8} - q^{10} - 5 q^{11} + 5 q^{14} + 11 q^{16} + q^{17} - 7 q^{19} - 15 q^{20} - 9 q^{22} + 11 q^{25} + 5 q^{28} + 2 q^{29} - 16 q^{31} - 22 q^{32} - 8 q^{34} - 4 q^{35} + 22 q^{37} + 3 q^{40} - 11 q^{41} - 15 q^{43} - 16 q^{44} - 7 q^{46} - 7 q^{47} - 15 q^{49} + 3 q^{50} + 17 q^{53} + 3 q^{55} - q^{56} + 12 q^{58} + 6 q^{59} - 13 q^{61} + 2 q^{62} + 11 q^{67} + 13 q^{68} - 24 q^{70} + 6 q^{73} - 15 q^{74} - 21 q^{76} - 15 q^{77} + 3 q^{79} + 20 q^{80} - 3 q^{82} - 12 q^{83} + 19 q^{85} + 29 q^{86} + 13 q^{88} - q^{89} - 7 q^{92} + 21 q^{95} - 5 q^{97} + 29 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.69202 −1.90355 −0.951773 0.306802i \(-0.900741\pi\)
−0.951773 + 0.306802i \(0.900741\pi\)
\(3\) 0 0
\(4\) 5.24698 2.62349
\(5\) 1.04892 0.469090 0.234545 0.972105i \(-0.424640\pi\)
0.234545 + 0.972105i \(0.424640\pi\)
\(6\) 0 0
\(7\) 0.554958 0.209754 0.104877 0.994485i \(-0.466555\pi\)
0.104877 + 0.994485i \(0.466555\pi\)
\(8\) −8.74094 −3.09039
\(9\) 0 0
\(10\) −2.82371 −0.892935
\(11\) −2.91185 −0.877957 −0.438979 0.898498i \(-0.644660\pi\)
−0.438979 + 0.898498i \(0.644660\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −1.49396 −0.399277
\(15\) 0 0
\(16\) 13.0368 3.25921
\(17\) 4.85086 1.17651 0.588253 0.808677i \(-0.299816\pi\)
0.588253 + 0.808677i \(0.299816\pi\)
\(18\) 0 0
\(19\) −0.753020 −0.172755 −0.0863774 0.996262i \(-0.527529\pi\)
−0.0863774 + 0.996262i \(0.527529\pi\)
\(20\) 5.50365 1.23065
\(21\) 0 0
\(22\) 7.83877 1.67123
\(23\) −5.76271 −1.20161 −0.600804 0.799396i \(-0.705153\pi\)
−0.600804 + 0.799396i \(0.705153\pi\)
\(24\) 0 0
\(25\) −3.89977 −0.779954
\(26\) 0 0
\(27\) 0 0
\(28\) 2.91185 0.550289
\(29\) 1.91185 0.355022 0.177511 0.984119i \(-0.443195\pi\)
0.177511 + 0.984119i \(0.443195\pi\)
\(30\) 0 0
\(31\) −9.51573 −1.70908 −0.854538 0.519389i \(-0.826159\pi\)
−0.854538 + 0.519389i \(0.826159\pi\)
\(32\) −17.6136 −3.11367
\(33\) 0 0
\(34\) −13.0586 −2.23953
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) 5.75302 0.945791 0.472895 0.881119i \(-0.343209\pi\)
0.472895 + 0.881119i \(0.343209\pi\)
\(38\) 2.02715 0.328847
\(39\) 0 0
\(40\) −9.16852 −1.44967
\(41\) −4.91185 −0.767103 −0.383551 0.923520i \(-0.625299\pi\)
−0.383551 + 0.923520i \(0.625299\pi\)
\(42\) 0 0
\(43\) −11.0978 −1.69240 −0.846202 0.532862i \(-0.821116\pi\)
−0.846202 + 0.532862i \(0.821116\pi\)
\(44\) −15.2784 −2.30331
\(45\) 0 0
\(46\) 15.5133 2.28732
\(47\) −0.753020 −0.109839 −0.0549197 0.998491i \(-0.517490\pi\)
−0.0549197 + 0.998491i \(0.517490\pi\)
\(48\) 0 0
\(49\) −6.69202 −0.956003
\(50\) 10.4983 1.48468
\(51\) 0 0
\(52\) 0 0
\(53\) 7.58211 1.04148 0.520741 0.853715i \(-0.325656\pi\)
0.520741 + 0.853715i \(0.325656\pi\)
\(54\) 0 0
\(55\) −3.05429 −0.411841
\(56\) −4.85086 −0.648223
\(57\) 0 0
\(58\) −5.14675 −0.675802
\(59\) −4.09783 −0.533493 −0.266746 0.963767i \(-0.585949\pi\)
−0.266746 + 0.963767i \(0.585949\pi\)
\(60\) 0 0
\(61\) −3.42327 −0.438305 −0.219153 0.975691i \(-0.570329\pi\)
−0.219153 + 0.975691i \(0.570329\pi\)
\(62\) 25.6165 3.25330
\(63\) 0 0
\(64\) 21.3424 2.66780
\(65\) 0 0
\(66\) 0 0
\(67\) −1.87263 −0.228778 −0.114389 0.993436i \(-0.536491\pi\)
−0.114389 + 0.993436i \(0.536491\pi\)
\(68\) 25.4523 3.08655
\(69\) 0 0
\(70\) −1.56704 −0.187297
\(71\) 10.5036 1.24655 0.623277 0.782001i \(-0.285801\pi\)
0.623277 + 0.782001i \(0.285801\pi\)
\(72\) 0 0
\(73\) 10.4765 1.22618 0.613091 0.790012i \(-0.289926\pi\)
0.613091 + 0.790012i \(0.289926\pi\)
\(74\) −15.4873 −1.80036
\(75\) 0 0
\(76\) −3.95108 −0.453220
\(77\) −1.61596 −0.184155
\(78\) 0 0
\(79\) 1.33513 0.150213 0.0751067 0.997176i \(-0.476070\pi\)
0.0751067 + 0.997176i \(0.476070\pi\)
\(80\) 13.6746 1.52886
\(81\) 0 0
\(82\) 13.2228 1.46022
\(83\) −2.64310 −0.290118 −0.145059 0.989423i \(-0.546337\pi\)
−0.145059 + 0.989423i \(0.546337\pi\)
\(84\) 0 0
\(85\) 5.08815 0.551887
\(86\) 29.8756 3.22157
\(87\) 0 0
\(88\) 25.4523 2.71323
\(89\) −9.92692 −1.05225 −0.526126 0.850407i \(-0.676356\pi\)
−0.526126 + 0.850407i \(0.676356\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −30.2368 −3.15241
\(93\) 0 0
\(94\) 2.02715 0.209084
\(95\) −0.789856 −0.0810375
\(96\) 0 0
\(97\) 17.0737 1.73357 0.866784 0.498683i \(-0.166183\pi\)
0.866784 + 0.498683i \(0.166183\pi\)
\(98\) 18.0151 1.81980
\(99\) 0 0
\(100\) −20.4620 −2.04620
\(101\) −7.32304 −0.728670 −0.364335 0.931268i \(-0.618704\pi\)
−0.364335 + 0.931268i \(0.618704\pi\)
\(102\) 0 0
\(103\) −4.21983 −0.415792 −0.207896 0.978151i \(-0.566662\pi\)
−0.207896 + 0.978151i \(0.566662\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −20.4112 −1.98251
\(107\) −6.39373 −0.618105 −0.309053 0.951045i \(-0.600012\pi\)
−0.309053 + 0.951045i \(0.600012\pi\)
\(108\) 0 0
\(109\) −3.46011 −0.331418 −0.165709 0.986175i \(-0.552991\pi\)
−0.165709 + 0.986175i \(0.552991\pi\)
\(110\) 8.22223 0.783958
\(111\) 0 0
\(112\) 7.23490 0.683634
\(113\) −9.35690 −0.880223 −0.440111 0.897943i \(-0.645061\pi\)
−0.440111 + 0.897943i \(0.645061\pi\)
\(114\) 0 0
\(115\) −6.04461 −0.563662
\(116\) 10.0315 0.931398
\(117\) 0 0
\(118\) 11.0315 1.01553
\(119\) 2.69202 0.246777
\(120\) 0 0
\(121\) −2.52111 −0.229191
\(122\) 9.21552 0.834334
\(123\) 0 0
\(124\) −49.9288 −4.48374
\(125\) −9.33513 −0.834959
\(126\) 0 0
\(127\) −4.48188 −0.397702 −0.198851 0.980030i \(-0.563721\pi\)
−0.198851 + 0.980030i \(0.563721\pi\)
\(128\) −22.2271 −1.96462
\(129\) 0 0
\(130\) 0 0
\(131\) 9.21744 0.805331 0.402666 0.915347i \(-0.368084\pi\)
0.402666 + 0.915347i \(0.368084\pi\)
\(132\) 0 0
\(133\) −0.417895 −0.0362361
\(134\) 5.04115 0.435489
\(135\) 0 0
\(136\) −42.4010 −3.63586
\(137\) 7.46980 0.638188 0.319094 0.947723i \(-0.396621\pi\)
0.319094 + 0.947723i \(0.396621\pi\)
\(138\) 0 0
\(139\) −17.9976 −1.52654 −0.763269 0.646081i \(-0.776407\pi\)
−0.763269 + 0.646081i \(0.776407\pi\)
\(140\) 3.05429 0.258135
\(141\) 0 0
\(142\) −28.2760 −2.37287
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00538 0.166537
\(146\) −28.2030 −2.33409
\(147\) 0 0
\(148\) 30.1860 2.48127
\(149\) −15.3351 −1.25630 −0.628151 0.778091i \(-0.716188\pi\)
−0.628151 + 0.778091i \(0.716188\pi\)
\(150\) 0 0
\(151\) 2.53079 0.205953 0.102977 0.994684i \(-0.467163\pi\)
0.102977 + 0.994684i \(0.467163\pi\)
\(152\) 6.58211 0.533879
\(153\) 0 0
\(154\) 4.35019 0.350548
\(155\) −9.98121 −0.801710
\(156\) 0 0
\(157\) 17.2392 1.37584 0.687919 0.725787i \(-0.258524\pi\)
0.687919 + 0.725787i \(0.258524\pi\)
\(158\) −3.59419 −0.285938
\(159\) 0 0
\(160\) −18.4752 −1.46059
\(161\) −3.19806 −0.252043
\(162\) 0 0
\(163\) 15.7071 1.23027 0.615137 0.788420i \(-0.289101\pi\)
0.615137 + 0.788420i \(0.289101\pi\)
\(164\) −25.7724 −2.01249
\(165\) 0 0
\(166\) 7.11529 0.552254
\(167\) −5.39612 −0.417565 −0.208782 0.977962i \(-0.566950\pi\)
−0.208782 + 0.977962i \(0.566950\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −13.6974 −1.05054
\(171\) 0 0
\(172\) −58.2301 −4.44000
\(173\) −23.9420 −1.82028 −0.910138 0.414306i \(-0.864024\pi\)
−0.910138 + 0.414306i \(0.864024\pi\)
\(174\) 0 0
\(175\) −2.16421 −0.163599
\(176\) −37.9614 −2.86145
\(177\) 0 0
\(178\) 26.7235 2.00301
\(179\) −18.4088 −1.37594 −0.687969 0.725740i \(-0.741498\pi\)
−0.687969 + 0.725740i \(0.741498\pi\)
\(180\) 0 0
\(181\) −3.63342 −0.270070 −0.135035 0.990841i \(-0.543115\pi\)
−0.135035 + 0.990841i \(0.543115\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 50.3715 3.71344
\(185\) 6.03444 0.443661
\(186\) 0 0
\(187\) −14.1250 −1.03292
\(188\) −3.95108 −0.288162
\(189\) 0 0
\(190\) 2.12631 0.154259
\(191\) 21.1782 1.53240 0.766201 0.642601i \(-0.222145\pi\)
0.766201 + 0.642601i \(0.222145\pi\)
\(192\) 0 0
\(193\) −17.6112 −1.26768 −0.633840 0.773464i \(-0.718522\pi\)
−0.633840 + 0.773464i \(0.718522\pi\)
\(194\) −45.9627 −3.29993
\(195\) 0 0
\(196\) −35.1129 −2.50806
\(197\) 4.66248 0.332188 0.166094 0.986110i \(-0.446884\pi\)
0.166094 + 0.986110i \(0.446884\pi\)
\(198\) 0 0
\(199\) 15.0368 1.06593 0.532967 0.846136i \(-0.321077\pi\)
0.532967 + 0.846136i \(0.321077\pi\)
\(200\) 34.0877 2.41036
\(201\) 0 0
\(202\) 19.7138 1.38706
\(203\) 1.06100 0.0744675
\(204\) 0 0
\(205\) −5.15213 −0.359840
\(206\) 11.3599 0.791480
\(207\) 0 0
\(208\) 0 0
\(209\) 2.19269 0.151671
\(210\) 0 0
\(211\) −0.460107 −0.0316751 −0.0158375 0.999875i \(-0.505041\pi\)
−0.0158375 + 0.999875i \(0.505041\pi\)
\(212\) 39.7832 2.73232
\(213\) 0 0
\(214\) 17.2121 1.17659
\(215\) −11.6407 −0.793890
\(216\) 0 0
\(217\) −5.28083 −0.358486
\(218\) 9.31468 0.630870
\(219\) 0 0
\(220\) −16.0258 −1.08046
\(221\) 0 0
\(222\) 0 0
\(223\) −16.3502 −1.09489 −0.547445 0.836842i \(-0.684399\pi\)
−0.547445 + 0.836842i \(0.684399\pi\)
\(224\) −9.77479 −0.653106
\(225\) 0 0
\(226\) 25.1890 1.67555
\(227\) −6.56033 −0.435425 −0.217712 0.976013i \(-0.569859\pi\)
−0.217712 + 0.976013i \(0.569859\pi\)
\(228\) 0 0
\(229\) −3.95539 −0.261380 −0.130690 0.991423i \(-0.541719\pi\)
−0.130690 + 0.991423i \(0.541719\pi\)
\(230\) 16.2722 1.07296
\(231\) 0 0
\(232\) −16.7114 −1.09716
\(233\) −8.35690 −0.547478 −0.273739 0.961804i \(-0.588261\pi\)
−0.273739 + 0.961804i \(0.588261\pi\)
\(234\) 0 0
\(235\) −0.789856 −0.0515245
\(236\) −21.5013 −1.39961
\(237\) 0 0
\(238\) −7.24698 −0.469752
\(239\) −20.1008 −1.30021 −0.650107 0.759843i \(-0.725276\pi\)
−0.650107 + 0.759843i \(0.725276\pi\)
\(240\) 0 0
\(241\) −19.0127 −1.22471 −0.612357 0.790581i \(-0.709778\pi\)
−0.612357 + 0.790581i \(0.709778\pi\)
\(242\) 6.78687 0.436277
\(243\) 0 0
\(244\) −17.9618 −1.14989
\(245\) −7.01938 −0.448452
\(246\) 0 0
\(247\) 0 0
\(248\) 83.1764 5.28171
\(249\) 0 0
\(250\) 25.1304 1.58938
\(251\) 0.763774 0.0482090 0.0241045 0.999709i \(-0.492327\pi\)
0.0241045 + 0.999709i \(0.492327\pi\)
\(252\) 0 0
\(253\) 16.7802 1.05496
\(254\) 12.0653 0.757045
\(255\) 0 0
\(256\) 17.1511 1.07194
\(257\) 13.0911 0.816602 0.408301 0.912847i \(-0.366121\pi\)
0.408301 + 0.912847i \(0.366121\pi\)
\(258\) 0 0
\(259\) 3.19269 0.198384
\(260\) 0 0
\(261\) 0 0
\(262\) −24.8135 −1.53299
\(263\) −18.3773 −1.13320 −0.566598 0.823995i \(-0.691741\pi\)
−0.566598 + 0.823995i \(0.691741\pi\)
\(264\) 0 0
\(265\) 7.95300 0.488549
\(266\) 1.12498 0.0689771
\(267\) 0 0
\(268\) −9.82563 −0.600196
\(269\) −23.6625 −1.44273 −0.721363 0.692557i \(-0.756484\pi\)
−0.721363 + 0.692557i \(0.756484\pi\)
\(270\) 0 0
\(271\) −19.7530 −1.19991 −0.599955 0.800034i \(-0.704815\pi\)
−0.599955 + 0.800034i \(0.704815\pi\)
\(272\) 63.2398 3.83448
\(273\) 0 0
\(274\) −20.1089 −1.21482
\(275\) 11.3556 0.684767
\(276\) 0 0
\(277\) 1.77777 0.106816 0.0534081 0.998573i \(-0.482992\pi\)
0.0534081 + 0.998573i \(0.482992\pi\)
\(278\) 48.4499 2.90583
\(279\) 0 0
\(280\) −5.08815 −0.304075
\(281\) −1.62133 −0.0967207 −0.0483603 0.998830i \(-0.515400\pi\)
−0.0483603 + 0.998830i \(0.515400\pi\)
\(282\) 0 0
\(283\) 4.98361 0.296245 0.148122 0.988969i \(-0.452677\pi\)
0.148122 + 0.988969i \(0.452677\pi\)
\(284\) 55.1124 3.27032
\(285\) 0 0
\(286\) 0 0
\(287\) −2.72587 −0.160903
\(288\) 0 0
\(289\) 6.53079 0.384164
\(290\) −5.39852 −0.317012
\(291\) 0 0
\(292\) 54.9700 3.21688
\(293\) 0.0717525 0.00419183 0.00209591 0.999998i \(-0.499333\pi\)
0.00209591 + 0.999998i \(0.499333\pi\)
\(294\) 0 0
\(295\) −4.29829 −0.250256
\(296\) −50.2868 −2.92286
\(297\) 0 0
\(298\) 41.2825 2.39143
\(299\) 0 0
\(300\) 0 0
\(301\) −6.15883 −0.354989
\(302\) −6.81295 −0.392041
\(303\) 0 0
\(304\) −9.81700 −0.563044
\(305\) −3.59073 −0.205605
\(306\) 0 0
\(307\) −5.19806 −0.296669 −0.148335 0.988937i \(-0.547391\pi\)
−0.148335 + 0.988937i \(0.547391\pi\)
\(308\) −8.47889 −0.483130
\(309\) 0 0
\(310\) 26.8696 1.52609
\(311\) 22.5429 1.27829 0.639145 0.769087i \(-0.279289\pi\)
0.639145 + 0.769087i \(0.279289\pi\)
\(312\) 0 0
\(313\) 22.6612 1.28088 0.640442 0.768006i \(-0.278751\pi\)
0.640442 + 0.768006i \(0.278751\pi\)
\(314\) −46.4083 −2.61897
\(315\) 0 0
\(316\) 7.00538 0.394083
\(317\) −26.3424 −1.47954 −0.739769 0.672861i \(-0.765065\pi\)
−0.739769 + 0.672861i \(0.765065\pi\)
\(318\) 0 0
\(319\) −5.56704 −0.311694
\(320\) 22.3864 1.25144
\(321\) 0 0
\(322\) 8.60925 0.479775
\(323\) −3.65279 −0.203247
\(324\) 0 0
\(325\) 0 0
\(326\) −42.2838 −2.34188
\(327\) 0 0
\(328\) 42.9342 2.37065
\(329\) −0.417895 −0.0230393
\(330\) 0 0
\(331\) −11.2295 −0.617230 −0.308615 0.951187i \(-0.599866\pi\)
−0.308615 + 0.951187i \(0.599866\pi\)
\(332\) −13.8683 −0.761123
\(333\) 0 0
\(334\) 14.5265 0.794854
\(335\) −1.96423 −0.107317
\(336\) 0 0
\(337\) 2.30798 0.125724 0.0628618 0.998022i \(-0.479977\pi\)
0.0628618 + 0.998022i \(0.479977\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 26.6974 1.44787
\(341\) 27.7084 1.50049
\(342\) 0 0
\(343\) −7.59850 −0.410280
\(344\) 97.0055 5.23019
\(345\) 0 0
\(346\) 64.4523 3.46498
\(347\) 9.13706 0.490503 0.245252 0.969459i \(-0.421129\pi\)
0.245252 + 0.969459i \(0.421129\pi\)
\(348\) 0 0
\(349\) 23.9758 1.28340 0.641699 0.766957i \(-0.278230\pi\)
0.641699 + 0.766957i \(0.278230\pi\)
\(350\) 5.82610 0.311418
\(351\) 0 0
\(352\) 51.2881 2.73367
\(353\) 27.1239 1.44366 0.721830 0.692070i \(-0.243301\pi\)
0.721830 + 0.692070i \(0.243301\pi\)
\(354\) 0 0
\(355\) 11.0175 0.584746
\(356\) −52.0863 −2.76057
\(357\) 0 0
\(358\) 49.5569 2.61916
\(359\) 26.0790 1.37640 0.688200 0.725521i \(-0.258401\pi\)
0.688200 + 0.725521i \(0.258401\pi\)
\(360\) 0 0
\(361\) −18.4330 −0.970156
\(362\) 9.78123 0.514090
\(363\) 0 0
\(364\) 0 0
\(365\) 10.9890 0.575190
\(366\) 0 0
\(367\) 9.57434 0.499776 0.249888 0.968275i \(-0.419606\pi\)
0.249888 + 0.968275i \(0.419606\pi\)
\(368\) −75.1275 −3.91629
\(369\) 0 0
\(370\) −16.2448 −0.844530
\(371\) 4.20775 0.218456
\(372\) 0 0
\(373\) 28.1497 1.45754 0.728769 0.684760i \(-0.240093\pi\)
0.728769 + 0.684760i \(0.240093\pi\)
\(374\) 38.0248 1.96621
\(375\) 0 0
\(376\) 6.58211 0.339446
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0465 0.824255 0.412127 0.911126i \(-0.364786\pi\)
0.412127 + 0.911126i \(0.364786\pi\)
\(380\) −4.14436 −0.212601
\(381\) 0 0
\(382\) −57.0122 −2.91700
\(383\) 24.6165 1.25785 0.628923 0.777467i \(-0.283496\pi\)
0.628923 + 0.777467i \(0.283496\pi\)
\(384\) 0 0
\(385\) −1.69501 −0.0863855
\(386\) 47.4097 2.41309
\(387\) 0 0
\(388\) 89.5852 4.54800
\(389\) 17.2198 0.873080 0.436540 0.899685i \(-0.356204\pi\)
0.436540 + 0.899685i \(0.356204\pi\)
\(390\) 0 0
\(391\) −27.9541 −1.41370
\(392\) 58.4946 2.95442
\(393\) 0 0
\(394\) −12.5515 −0.632335
\(395\) 1.40044 0.0704636
\(396\) 0 0
\(397\) −2.03923 −0.102346 −0.0511730 0.998690i \(-0.516296\pi\)
−0.0511730 + 0.998690i \(0.516296\pi\)
\(398\) −40.4795 −2.02905
\(399\) 0 0
\(400\) −50.8407 −2.54203
\(401\) 1.46144 0.0729806 0.0364903 0.999334i \(-0.488382\pi\)
0.0364903 + 0.999334i \(0.488382\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −38.4239 −1.91166
\(405\) 0 0
\(406\) −2.85623 −0.141752
\(407\) −16.7520 −0.830364
\(408\) 0 0
\(409\) −29.9390 −1.48039 −0.740194 0.672393i \(-0.765266\pi\)
−0.740194 + 0.672393i \(0.765266\pi\)
\(410\) 13.8696 0.684973
\(411\) 0 0
\(412\) −22.1414 −1.09083
\(413\) −2.27413 −0.111902
\(414\) 0 0
\(415\) −2.77240 −0.136092
\(416\) 0 0
\(417\) 0 0
\(418\) −5.90276 −0.288713
\(419\) −6.64742 −0.324748 −0.162374 0.986729i \(-0.551915\pi\)
−0.162374 + 0.986729i \(0.551915\pi\)
\(420\) 0 0
\(421\) 13.5646 0.661100 0.330550 0.943788i \(-0.392766\pi\)
0.330550 + 0.943788i \(0.392766\pi\)
\(422\) 1.23862 0.0602950
\(423\) 0 0
\(424\) −66.2747 −3.21858
\(425\) −18.9172 −0.917620
\(426\) 0 0
\(427\) −1.89977 −0.0919364
\(428\) −33.5478 −1.62159
\(429\) 0 0
\(430\) 31.3370 1.51121
\(431\) −35.9463 −1.73147 −0.865736 0.500501i \(-0.833149\pi\)
−0.865736 + 0.500501i \(0.833149\pi\)
\(432\) 0 0
\(433\) −32.4741 −1.56061 −0.780303 0.625402i \(-0.784935\pi\)
−0.780303 + 0.625402i \(0.784935\pi\)
\(434\) 14.2161 0.682395
\(435\) 0 0
\(436\) −18.1551 −0.869472
\(437\) 4.33944 0.207583
\(438\) 0 0
\(439\) 12.8321 0.612441 0.306221 0.951961i \(-0.400935\pi\)
0.306221 + 0.951961i \(0.400935\pi\)
\(440\) 26.6974 1.27275
\(441\) 0 0
\(442\) 0 0
\(443\) −11.9608 −0.568273 −0.284137 0.958784i \(-0.591707\pi\)
−0.284137 + 0.958784i \(0.591707\pi\)
\(444\) 0 0
\(445\) −10.4125 −0.493601
\(446\) 44.0151 2.08417
\(447\) 0 0
\(448\) 11.8442 0.559584
\(449\) 12.4789 0.588915 0.294458 0.955665i \(-0.404861\pi\)
0.294458 + 0.955665i \(0.404861\pi\)
\(450\) 0 0
\(451\) 14.3026 0.673483
\(452\) −49.0954 −2.30926
\(453\) 0 0
\(454\) 17.6606 0.828851
\(455\) 0 0
\(456\) 0 0
\(457\) 32.4523 1.51806 0.759028 0.651058i \(-0.225674\pi\)
0.759028 + 0.651058i \(0.225674\pi\)
\(458\) 10.6480 0.497549
\(459\) 0 0
\(460\) −31.7159 −1.47876
\(461\) −24.4034 −1.13658 −0.568290 0.822828i \(-0.692395\pi\)
−0.568290 + 0.822828i \(0.692395\pi\)
\(462\) 0 0
\(463\) −33.1836 −1.54217 −0.771086 0.636731i \(-0.780286\pi\)
−0.771086 + 0.636731i \(0.780286\pi\)
\(464\) 24.9245 1.15709
\(465\) 0 0
\(466\) 22.4969 1.04215
\(467\) 38.5206 1.78252 0.891261 0.453490i \(-0.149821\pi\)
0.891261 + 0.453490i \(0.149821\pi\)
\(468\) 0 0
\(469\) −1.03923 −0.0479871
\(470\) 2.12631 0.0980794
\(471\) 0 0
\(472\) 35.8189 1.64870
\(473\) 32.3153 1.48586
\(474\) 0 0
\(475\) 2.93661 0.134741
\(476\) 14.1250 0.647417
\(477\) 0 0
\(478\) 54.1118 2.47502
\(479\) 8.34481 0.381284 0.190642 0.981660i \(-0.438943\pi\)
0.190642 + 0.981660i \(0.438943\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 51.1825 2.33130
\(483\) 0 0
\(484\) −13.2282 −0.601282
\(485\) 17.9089 0.813200
\(486\) 0 0
\(487\) −14.8586 −0.673309 −0.336654 0.941628i \(-0.609295\pi\)
−0.336654 + 0.941628i \(0.609295\pi\)
\(488\) 29.9226 1.35453
\(489\) 0 0
\(490\) 18.8963 0.853648
\(491\) −4.99894 −0.225599 −0.112799 0.993618i \(-0.535982\pi\)
−0.112799 + 0.993618i \(0.535982\pi\)
\(492\) 0 0
\(493\) 9.27413 0.417686
\(494\) 0 0
\(495\) 0 0
\(496\) −124.055 −5.57023
\(497\) 5.82908 0.261470
\(498\) 0 0
\(499\) −0.385371 −0.0172516 −0.00862579 0.999963i \(-0.502746\pi\)
−0.00862579 + 0.999963i \(0.502746\pi\)
\(500\) −48.9812 −2.19051
\(501\) 0 0
\(502\) −2.05610 −0.0917681
\(503\) −22.6179 −1.00848 −0.504241 0.863563i \(-0.668228\pi\)
−0.504241 + 0.863563i \(0.668228\pi\)
\(504\) 0 0
\(505\) −7.68127 −0.341812
\(506\) −45.1726 −2.00817
\(507\) 0 0
\(508\) −23.5163 −1.04337
\(509\) 11.6039 0.514333 0.257166 0.966367i \(-0.417211\pi\)
0.257166 + 0.966367i \(0.417211\pi\)
\(510\) 0 0
\(511\) 5.81402 0.257197
\(512\) −1.71678 −0.0758715
\(513\) 0 0
\(514\) −35.2416 −1.55444
\(515\) −4.42626 −0.195044
\(516\) 0 0
\(517\) 2.19269 0.0964342
\(518\) −8.59478 −0.377633
\(519\) 0 0
\(520\) 0 0
\(521\) 1.62671 0.0712675 0.0356337 0.999365i \(-0.488655\pi\)
0.0356337 + 0.999365i \(0.488655\pi\)
\(522\) 0 0
\(523\) 10.0718 0.440407 0.220203 0.975454i \(-0.429328\pi\)
0.220203 + 0.975454i \(0.429328\pi\)
\(524\) 48.3637 2.11278
\(525\) 0 0
\(526\) 49.4722 2.15709
\(527\) −46.1594 −2.01074
\(528\) 0 0
\(529\) 10.2088 0.443862
\(530\) −21.4097 −0.929976
\(531\) 0 0
\(532\) −2.19269 −0.0950650
\(533\) 0 0
\(534\) 0 0
\(535\) −6.70650 −0.289947
\(536\) 16.3685 0.707012
\(537\) 0 0
\(538\) 63.6999 2.74630
\(539\) 19.4862 0.839330
\(540\) 0 0
\(541\) −20.4674 −0.879962 −0.439981 0.898007i \(-0.645015\pi\)
−0.439981 + 0.898007i \(0.645015\pi\)
\(542\) 53.1756 2.28409
\(543\) 0 0
\(544\) −85.4408 −3.66325
\(545\) −3.62937 −0.155465
\(546\) 0 0
\(547\) −27.5478 −1.17786 −0.588929 0.808185i \(-0.700450\pi\)
−0.588929 + 0.808185i \(0.700450\pi\)
\(548\) 39.1939 1.67428
\(549\) 0 0
\(550\) −30.5694 −1.30348
\(551\) −1.43967 −0.0613318
\(552\) 0 0
\(553\) 0.740939 0.0315079
\(554\) −4.78581 −0.203329
\(555\) 0 0
\(556\) −94.4331 −4.00485
\(557\) −37.9855 −1.60950 −0.804749 0.593615i \(-0.797700\pi\)
−0.804749 + 0.593615i \(0.797700\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 7.58881 0.320686
\(561\) 0 0
\(562\) 4.36467 0.184112
\(563\) 29.7724 1.25476 0.627378 0.778714i \(-0.284128\pi\)
0.627378 + 0.778714i \(0.284128\pi\)
\(564\) 0 0
\(565\) −9.81461 −0.412904
\(566\) −13.4160 −0.563916
\(567\) 0 0
\(568\) −91.8117 −3.85234
\(569\) 21.9541 0.920362 0.460181 0.887825i \(-0.347784\pi\)
0.460181 + 0.887825i \(0.347784\pi\)
\(570\) 0 0
\(571\) 2.46575 0.103188 0.0515942 0.998668i \(-0.483570\pi\)
0.0515942 + 0.998668i \(0.483570\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 7.33811 0.306287
\(575\) 22.4733 0.937199
\(576\) 0 0
\(577\) 17.4547 0.726650 0.363325 0.931662i \(-0.381641\pi\)
0.363325 + 0.931662i \(0.381641\pi\)
\(578\) −17.5810 −0.731275
\(579\) 0 0
\(580\) 10.5222 0.436909
\(581\) −1.46681 −0.0608536
\(582\) 0 0
\(583\) −22.0780 −0.914377
\(584\) −91.5745 −3.78938
\(585\) 0 0
\(586\) −0.193159 −0.00797934
\(587\) −6.26337 −0.258517 −0.129259 0.991611i \(-0.541260\pi\)
−0.129259 + 0.991611i \(0.541260\pi\)
\(588\) 0 0
\(589\) 7.16554 0.295251
\(590\) 11.5711 0.476374
\(591\) 0 0
\(592\) 75.0012 3.08253
\(593\) −22.8745 −0.939345 −0.469672 0.882841i \(-0.655628\pi\)
−0.469672 + 0.882841i \(0.655628\pi\)
\(594\) 0 0
\(595\) 2.82371 0.115761
\(596\) −80.4631 −3.29590
\(597\) 0 0
\(598\) 0 0
\(599\) 1.05621 0.0431557 0.0215778 0.999767i \(-0.493131\pi\)
0.0215778 + 0.999767i \(0.493131\pi\)
\(600\) 0 0
\(601\) −33.3236 −1.35930 −0.679650 0.733537i \(-0.737868\pi\)
−0.679650 + 0.733537i \(0.737868\pi\)
\(602\) 16.5797 0.675739
\(603\) 0 0
\(604\) 13.2790 0.540316
\(605\) −2.64443 −0.107511
\(606\) 0 0
\(607\) 16.2403 0.659172 0.329586 0.944125i \(-0.393091\pi\)
0.329586 + 0.944125i \(0.393091\pi\)
\(608\) 13.2634 0.537901
\(609\) 0 0
\(610\) 9.66632 0.391378
\(611\) 0 0
\(612\) 0 0
\(613\) 11.0479 0.446219 0.223109 0.974793i \(-0.428379\pi\)
0.223109 + 0.974793i \(0.428379\pi\)
\(614\) 13.9933 0.564723
\(615\) 0 0
\(616\) 14.1250 0.569112
\(617\) 4.65950 0.187584 0.0937922 0.995592i \(-0.470101\pi\)
0.0937922 + 0.995592i \(0.470101\pi\)
\(618\) 0 0
\(619\) 31.9259 1.28321 0.641604 0.767036i \(-0.278269\pi\)
0.641604 + 0.767036i \(0.278269\pi\)
\(620\) −52.3712 −2.10328
\(621\) 0 0
\(622\) −60.6859 −2.43328
\(623\) −5.50902 −0.220714
\(624\) 0 0
\(625\) 9.70709 0.388283
\(626\) −61.0043 −2.43822
\(627\) 0 0
\(628\) 90.4538 3.60950
\(629\) 27.9071 1.11273
\(630\) 0 0
\(631\) −39.4413 −1.57013 −0.785067 0.619411i \(-0.787372\pi\)
−0.785067 + 0.619411i \(0.787372\pi\)
\(632\) −11.6703 −0.464218
\(633\) 0 0
\(634\) 70.9144 2.81637
\(635\) −4.70112 −0.186558
\(636\) 0 0
\(637\) 0 0
\(638\) 14.9866 0.593325
\(639\) 0 0
\(640\) −23.3144 −0.921583
\(641\) −45.4510 −1.79521 −0.897603 0.440804i \(-0.854693\pi\)
−0.897603 + 0.440804i \(0.854693\pi\)
\(642\) 0 0
\(643\) −29.7469 −1.17310 −0.586552 0.809912i \(-0.699515\pi\)
−0.586552 + 0.809912i \(0.699515\pi\)
\(644\) −16.7802 −0.661231
\(645\) 0 0
\(646\) 9.83340 0.386890
\(647\) 16.9312 0.665635 0.332818 0.942991i \(-0.392001\pi\)
0.332818 + 0.942991i \(0.392001\pi\)
\(648\) 0 0
\(649\) 11.9323 0.468384
\(650\) 0 0
\(651\) 0 0
\(652\) 82.4148 3.22761
\(653\) 33.1976 1.29912 0.649561 0.760309i \(-0.274953\pi\)
0.649561 + 0.760309i \(0.274953\pi\)
\(654\) 0 0
\(655\) 9.66833 0.377773
\(656\) −64.0350 −2.50015
\(657\) 0 0
\(658\) 1.12498 0.0438564
\(659\) 42.6571 1.66168 0.830842 0.556508i \(-0.187859\pi\)
0.830842 + 0.556508i \(0.187859\pi\)
\(660\) 0 0
\(661\) 38.6902 1.50488 0.752438 0.658664i \(-0.228878\pi\)
0.752438 + 0.658664i \(0.228878\pi\)
\(662\) 30.2301 1.17493
\(663\) 0 0
\(664\) 23.1032 0.896578
\(665\) −0.438337 −0.0169980
\(666\) 0 0
\(667\) −11.0175 −0.426598
\(668\) −28.3134 −1.09548
\(669\) 0 0
\(670\) 5.28775 0.204283
\(671\) 9.96807 0.384813
\(672\) 0 0
\(673\) −2.59419 −0.0999986 −0.0499993 0.998749i \(-0.515922\pi\)
−0.0499993 + 0.998749i \(0.515922\pi\)
\(674\) −6.21313 −0.239321
\(675\) 0 0
\(676\) 0 0
\(677\) −1.75302 −0.0673740 −0.0336870 0.999432i \(-0.510725\pi\)
−0.0336870 + 0.999432i \(0.510725\pi\)
\(678\) 0 0
\(679\) 9.47517 0.363624
\(680\) −44.4752 −1.70555
\(681\) 0 0
\(682\) −74.5916 −2.85626
\(683\) −16.3351 −0.625046 −0.312523 0.949910i \(-0.601174\pi\)
−0.312523 + 0.949910i \(0.601174\pi\)
\(684\) 0 0
\(685\) 7.83520 0.299368
\(686\) 20.4553 0.780988
\(687\) 0 0
\(688\) −144.681 −5.51590
\(689\) 0 0
\(690\) 0 0
\(691\) 15.9105 0.605265 0.302632 0.953107i \(-0.402135\pi\)
0.302632 + 0.953107i \(0.402135\pi\)
\(692\) −125.623 −4.77547
\(693\) 0 0
\(694\) −24.5972 −0.933696
\(695\) −18.8780 −0.716083
\(696\) 0 0
\(697\) −23.8267 −0.902500
\(698\) −64.5435 −2.44301
\(699\) 0 0
\(700\) −11.3556 −0.429200
\(701\) 20.8635 0.788005 0.394002 0.919109i \(-0.371090\pi\)
0.394002 + 0.919109i \(0.371090\pi\)
\(702\) 0 0
\(703\) −4.33214 −0.163390
\(704\) −62.1460 −2.34222
\(705\) 0 0
\(706\) −73.0182 −2.74807
\(707\) −4.06398 −0.152842
\(708\) 0 0
\(709\) 15.6485 0.587691 0.293846 0.955853i \(-0.405065\pi\)
0.293846 + 0.955853i \(0.405065\pi\)
\(710\) −29.6592 −1.11309
\(711\) 0 0
\(712\) 86.7706 3.25187
\(713\) 54.8364 2.05364
\(714\) 0 0
\(715\) 0 0
\(716\) −96.5906 −3.60976
\(717\) 0 0
\(718\) −70.2054 −2.62004
\(719\) 27.1594 1.01288 0.506438 0.862276i \(-0.330962\pi\)
0.506438 + 0.862276i \(0.330962\pi\)
\(720\) 0 0
\(721\) −2.34183 −0.0872143
\(722\) 49.6219 1.84674
\(723\) 0 0
\(724\) −19.0645 −0.708525
\(725\) −7.45580 −0.276901
\(726\) 0 0
\(727\) 31.7784 1.17859 0.589297 0.807916i \(-0.299405\pi\)
0.589297 + 0.807916i \(0.299405\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −29.5826 −1.09490
\(731\) −53.8340 −1.99112
\(732\) 0 0
\(733\) 46.8907 1.73195 0.865973 0.500090i \(-0.166700\pi\)
0.865973 + 0.500090i \(0.166700\pi\)
\(734\) −25.7743 −0.951347
\(735\) 0 0
\(736\) 101.502 3.74141
\(737\) 5.45281 0.200857
\(738\) 0 0
\(739\) 17.0479 0.627115 0.313558 0.949569i \(-0.398479\pi\)
0.313558 + 0.949569i \(0.398479\pi\)
\(740\) 31.6626 1.16394
\(741\) 0 0
\(742\) −11.3274 −0.415840
\(743\) 11.6324 0.426750 0.213375 0.976970i \(-0.431554\pi\)
0.213375 + 0.976970i \(0.431554\pi\)
\(744\) 0 0
\(745\) −16.0853 −0.589319
\(746\) −75.7797 −2.77449
\(747\) 0 0
\(748\) −74.1135 −2.70986
\(749\) −3.54825 −0.129650
\(750\) 0 0
\(751\) −13.0295 −0.475455 −0.237727 0.971332i \(-0.576402\pi\)
−0.237727 + 0.971332i \(0.576402\pi\)
\(752\) −9.81700 −0.357989
\(753\) 0 0
\(754\) 0 0
\(755\) 2.65459 0.0966106
\(756\) 0 0
\(757\) 22.7899 0.828311 0.414156 0.910206i \(-0.364077\pi\)
0.414156 + 0.910206i \(0.364077\pi\)
\(758\) −43.1976 −1.56901
\(759\) 0 0
\(760\) 6.90408 0.250437
\(761\) 38.3424 1.38991 0.694956 0.719052i \(-0.255424\pi\)
0.694956 + 0.719052i \(0.255424\pi\)
\(762\) 0 0
\(763\) −1.92021 −0.0695164
\(764\) 111.122 4.02024
\(765\) 0 0
\(766\) −66.2683 −2.39437
\(767\) 0 0
\(768\) 0 0
\(769\) −3.63879 −0.131218 −0.0656091 0.997845i \(-0.520899\pi\)
−0.0656091 + 0.997845i \(0.520899\pi\)
\(770\) 4.56299 0.164439
\(771\) 0 0
\(772\) −92.4055 −3.32575
\(773\) 39.3424 1.41505 0.707524 0.706689i \(-0.249812\pi\)
0.707524 + 0.706689i \(0.249812\pi\)
\(774\) 0 0
\(775\) 37.1092 1.33300
\(776\) −149.240 −5.35740
\(777\) 0 0
\(778\) −46.3562 −1.66195
\(779\) 3.69873 0.132521
\(780\) 0 0
\(781\) −30.5851 −1.09442
\(782\) 75.2529 2.69104
\(783\) 0 0
\(784\) −87.2428 −3.11581
\(785\) 18.0825 0.645392
\(786\) 0 0
\(787\) −25.4252 −0.906310 −0.453155 0.891432i \(-0.649702\pi\)
−0.453155 + 0.891432i \(0.649702\pi\)
\(788\) 24.4639 0.871492
\(789\) 0 0
\(790\) −3.77000 −0.134131
\(791\) −5.19269 −0.184631
\(792\) 0 0
\(793\) 0 0
\(794\) 5.48965 0.194820
\(795\) 0 0
\(796\) 78.8980 2.79646
\(797\) −20.7138 −0.733720 −0.366860 0.930276i \(-0.619567\pi\)
−0.366860 + 0.930276i \(0.619567\pi\)
\(798\) 0 0
\(799\) −3.65279 −0.129227
\(800\) 68.6889 2.42852
\(801\) 0 0
\(802\) −3.93422 −0.138922
\(803\) −30.5060 −1.07653
\(804\) 0 0
\(805\) −3.35450 −0.118231
\(806\) 0 0
\(807\) 0 0
\(808\) 64.0103 2.25187
\(809\) −11.0978 −0.390179 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(810\) 0 0
\(811\) −4.84223 −0.170034 −0.0850169 0.996380i \(-0.527094\pi\)
−0.0850169 + 0.996380i \(0.527094\pi\)
\(812\) 5.56704 0.195365
\(813\) 0 0
\(814\) 45.0966 1.58064
\(815\) 16.4754 0.577109
\(816\) 0 0
\(817\) 8.35690 0.292371
\(818\) 80.5964 2.81799
\(819\) 0 0
\(820\) −27.0331 −0.944037
\(821\) 43.7381 1.52647 0.763235 0.646121i \(-0.223610\pi\)
0.763235 + 0.646121i \(0.223610\pi\)
\(822\) 0 0
\(823\) 12.0998 0.421771 0.210885 0.977511i \(-0.432365\pi\)
0.210885 + 0.977511i \(0.432365\pi\)
\(824\) 36.8853 1.28496
\(825\) 0 0
\(826\) 6.12200 0.213012
\(827\) 35.3212 1.22824 0.614120 0.789213i \(-0.289511\pi\)
0.614120 + 0.789213i \(0.289511\pi\)
\(828\) 0 0
\(829\) 53.4946 1.85794 0.928971 0.370152i \(-0.120694\pi\)
0.928971 + 0.370152i \(0.120694\pi\)
\(830\) 7.46335 0.259057
\(831\) 0 0
\(832\) 0 0
\(833\) −32.4620 −1.12474
\(834\) 0 0
\(835\) −5.66009 −0.195875
\(836\) 11.5050 0.397908
\(837\) 0 0
\(838\) 17.8950 0.618172
\(839\) −23.1521 −0.799300 −0.399650 0.916668i \(-0.630868\pi\)
−0.399650 + 0.916668i \(0.630868\pi\)
\(840\) 0 0
\(841\) −25.3448 −0.873959
\(842\) −36.5163 −1.25844
\(843\) 0 0
\(844\) −2.41417 −0.0830993
\(845\) 0 0
\(846\) 0 0
\(847\) −1.39911 −0.0480739
\(848\) 98.8467 3.39441
\(849\) 0 0
\(850\) 50.9256 1.74673
\(851\) −33.1530 −1.13647
\(852\) 0 0
\(853\) 26.7265 0.915097 0.457548 0.889185i \(-0.348728\pi\)
0.457548 + 0.889185i \(0.348728\pi\)
\(854\) 5.11423 0.175005
\(855\) 0 0
\(856\) 55.8872 1.91019
\(857\) −42.6064 −1.45541 −0.727703 0.685892i \(-0.759412\pi\)
−0.727703 + 0.685892i \(0.759412\pi\)
\(858\) 0 0
\(859\) 33.6079 1.14669 0.573344 0.819315i \(-0.305646\pi\)
0.573344 + 0.819315i \(0.305646\pi\)
\(860\) −61.0786 −2.08276
\(861\) 0 0
\(862\) 96.7682 3.29594
\(863\) −18.7047 −0.636715 −0.318358 0.947971i \(-0.603131\pi\)
−0.318358 + 0.947971i \(0.603131\pi\)
\(864\) 0 0
\(865\) −25.1132 −0.853873
\(866\) 87.4210 2.97069
\(867\) 0 0
\(868\) −27.7084 −0.940485
\(869\) −3.88769 −0.131881
\(870\) 0 0
\(871\) 0 0
\(872\) 30.2446 1.02421
\(873\) 0 0
\(874\) −11.6819 −0.395145
\(875\) −5.18060 −0.175136
\(876\) 0 0
\(877\) 36.8237 1.24345 0.621724 0.783236i \(-0.286433\pi\)
0.621724 + 0.783236i \(0.286433\pi\)
\(878\) −34.5442 −1.16581
\(879\) 0 0
\(880\) −39.8183 −1.34228
\(881\) 41.1250 1.38554 0.692768 0.721161i \(-0.256391\pi\)
0.692768 + 0.721161i \(0.256391\pi\)
\(882\) 0 0
\(883\) 30.7482 1.03476 0.517380 0.855756i \(-0.326907\pi\)
0.517380 + 0.855756i \(0.326907\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 32.1987 1.08173
\(887\) −7.58940 −0.254827 −0.127414 0.991850i \(-0.540668\pi\)
−0.127414 + 0.991850i \(0.540668\pi\)
\(888\) 0 0
\(889\) −2.48725 −0.0834198
\(890\) 28.0307 0.939592
\(891\) 0 0
\(892\) −85.7891 −2.87243
\(893\) 0.567040 0.0189753
\(894\) 0 0
\(895\) −19.3093 −0.645439
\(896\) −12.3351 −0.412088
\(897\) 0 0
\(898\) −33.5934 −1.12103
\(899\) −18.1927 −0.606760
\(900\) 0 0
\(901\) 36.7797 1.22531
\(902\) −38.5029 −1.28201
\(903\) 0 0
\(904\) 81.7881 2.72023
\(905\) −3.81115 −0.126687
\(906\) 0 0
\(907\) 19.3333 0.641952 0.320976 0.947087i \(-0.395989\pi\)
0.320976 + 0.947087i \(0.395989\pi\)
\(908\) −34.4219 −1.14233
\(909\) 0 0
\(910\) 0 0
\(911\) −22.7149 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(912\) 0 0
\(913\) 7.69633 0.254711
\(914\) −87.3624 −2.88969
\(915\) 0 0
\(916\) −20.7539 −0.685727
\(917\) 5.11529 0.168922
\(918\) 0 0
\(919\) 14.2911 0.471420 0.235710 0.971823i \(-0.424258\pi\)
0.235710 + 0.971823i \(0.424258\pi\)
\(920\) 52.8355 1.74194
\(921\) 0 0
\(922\) 65.6945 2.16353
\(923\) 0 0
\(924\) 0 0
\(925\) −22.4355 −0.737674
\(926\) 89.3309 2.93560
\(927\) 0 0
\(928\) −33.6746 −1.10542
\(929\) −32.7211 −1.07354 −0.536772 0.843727i \(-0.680356\pi\)
−0.536772 + 0.843727i \(0.680356\pi\)
\(930\) 0 0
\(931\) 5.03923 0.165154
\(932\) −43.8485 −1.43630
\(933\) 0 0
\(934\) −103.698 −3.39311
\(935\) −14.8159 −0.484533
\(936\) 0 0
\(937\) −4.01400 −0.131132 −0.0655658 0.997848i \(-0.520885\pi\)
−0.0655658 + 0.997848i \(0.520885\pi\)
\(938\) 2.79763 0.0913457
\(939\) 0 0
\(940\) −4.14436 −0.135174
\(941\) 28.2669 0.921476 0.460738 0.887536i \(-0.347585\pi\)
0.460738 + 0.887536i \(0.347585\pi\)
\(942\) 0 0
\(943\) 28.3056 0.921757
\(944\) −53.4228 −1.73876
\(945\) 0 0
\(946\) −86.9934 −2.82840
\(947\) 37.8455 1.22981 0.614906 0.788600i \(-0.289194\pi\)
0.614906 + 0.788600i \(0.289194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −7.90541 −0.256485
\(951\) 0 0
\(952\) −23.5308 −0.762637
\(953\) 40.8256 1.32247 0.661236 0.750178i \(-0.270032\pi\)
0.661236 + 0.750178i \(0.270032\pi\)
\(954\) 0 0
\(955\) 22.2142 0.718834
\(956\) −105.469 −3.41110
\(957\) 0 0
\(958\) −22.4644 −0.725792
\(959\) 4.14542 0.133863
\(960\) 0 0
\(961\) 59.5491 1.92094
\(962\) 0 0
\(963\) 0 0
\(964\) −99.7591 −3.21302
\(965\) −18.4727 −0.594656
\(966\) 0 0
\(967\) 10.2798 0.330575 0.165288 0.986245i \(-0.447145\pi\)
0.165288 + 0.986245i \(0.447145\pi\)
\(968\) 22.0368 0.708291
\(969\) 0 0
\(970\) −48.2111 −1.54796
\(971\) 19.4805 0.625161 0.312580 0.949891i \(-0.398807\pi\)
0.312580 + 0.949891i \(0.398807\pi\)
\(972\) 0 0
\(973\) −9.98792 −0.320198
\(974\) 39.9997 1.28167
\(975\) 0 0
\(976\) −44.6286 −1.42853
\(977\) −47.0538 −1.50539 −0.752693 0.658372i \(-0.771245\pi\)
−0.752693 + 0.658372i \(0.771245\pi\)
\(978\) 0 0
\(979\) 28.9057 0.923831
\(980\) −36.8305 −1.17651
\(981\) 0 0
\(982\) 13.4572 0.429438
\(983\) −34.4295 −1.09813 −0.549065 0.835779i \(-0.685016\pi\)
−0.549065 + 0.835779i \(0.685016\pi\)
\(984\) 0 0
\(985\) 4.89056 0.155826
\(986\) −24.9661 −0.795084
\(987\) 0 0
\(988\) 0 0
\(989\) 63.9536 2.03361
\(990\) 0 0
\(991\) 7.30798 0.232146 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(992\) 167.606 5.32149
\(993\) 0 0
\(994\) −15.6920 −0.497721
\(995\) 15.7724 0.500019
\(996\) 0 0
\(997\) −35.6915 −1.13036 −0.565181 0.824967i \(-0.691194\pi\)
−0.565181 + 0.824967i \(0.691194\pi\)
\(998\) 1.03743 0.0328392
\(999\) 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 1521.2.a.n.1.1 3
3.2 odd 2 507.2.a.l.1.3 yes 3
12.11 even 2 8112.2.a.cp.1.1 3
13.5 odd 4 1521.2.b.k.1351.6 6
13.8 odd 4 1521.2.b.k.1351.1 6
13.12 even 2 1521.2.a.s.1.3 3
39.2 even 12 507.2.j.i.316.6 12
39.5 even 4 507.2.b.f.337.1 6
39.8 even 4 507.2.b.f.337.6 6
39.11 even 12 507.2.j.i.316.1 12
39.17 odd 6 507.2.e.l.484.3 6
39.20 even 12 507.2.j.i.361.6 12
39.23 odd 6 507.2.e.l.22.3 6
39.29 odd 6 507.2.e.i.22.1 6
39.32 even 12 507.2.j.i.361.1 12
39.35 odd 6 507.2.e.i.484.1 6
39.38 odd 2 507.2.a.i.1.1 3
156.155 even 2 8112.2.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.1 3 39.38 odd 2
507.2.a.l.1.3 yes 3 3.2 odd 2
507.2.b.f.337.1 6 39.5 even 4
507.2.b.f.337.6 6 39.8 even 4
507.2.e.i.22.1 6 39.29 odd 6
507.2.e.i.484.1 6 39.35 odd 6
507.2.e.l.22.3 6 39.23 odd 6
507.2.e.l.484.3 6 39.17 odd 6
507.2.j.i.316.1 12 39.11 even 12
507.2.j.i.316.6 12 39.2 even 12
507.2.j.i.361.1 12 39.32 even 12
507.2.j.i.361.6 12 39.20 even 12
1521.2.a.n.1.1 3 1.1 even 1 trivial
1521.2.a.s.1.3 3 13.12 even 2
1521.2.b.k.1351.1 6 13.8 odd 4
1521.2.b.k.1351.6 6 13.5 odd 4
8112.2.a.cg.1.3 3 156.155 even 2
8112.2.a.cp.1.1 3 12.11 even 2