Properties

Label 1521.2.a.m.1.2
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +4.56155 q^{4} +0.561553 q^{5} +3.56155 q^{7} +6.56155 q^{8} +O(q^{10})\) \(q+2.56155 q^{2} +4.56155 q^{4} +0.561553 q^{5} +3.56155 q^{7} +6.56155 q^{8} +1.43845 q^{10} -2.00000 q^{11} +9.12311 q^{14} +7.68466 q^{16} -2.56155 q^{17} +1.12311 q^{19} +2.56155 q^{20} -5.12311 q^{22} -2.00000 q^{23} -4.68466 q^{25} +16.2462 q^{28} +5.68466 q^{29} +1.56155 q^{31} +6.56155 q^{32} -6.56155 q^{34} +2.00000 q^{35} -3.43845 q^{37} +2.87689 q^{38} +3.68466 q^{40} +2.56155 q^{41} +0.438447 q^{43} -9.12311 q^{44} -5.12311 q^{46} -8.24621 q^{47} +5.68466 q^{49} -12.0000 q^{50} -11.6847 q^{53} -1.12311 q^{55} +23.3693 q^{56} +14.5616 q^{58} -11.1231 q^{59} +12.1231 q^{61} +4.00000 q^{62} +1.43845 q^{64} -0.438447 q^{67} -11.6847 q^{68} +5.12311 q^{70} +14.0000 q^{71} +1.87689 q^{73} -8.80776 q^{74} +5.12311 q^{76} -7.12311 q^{77} +9.56155 q^{79} +4.31534 q^{80} +6.56155 q^{82} -9.12311 q^{83} -1.43845 q^{85} +1.12311 q^{86} -13.1231 q^{88} +13.1231 q^{89} -9.12311 q^{92} -21.1231 q^{94} +0.630683 q^{95} +4.43845 q^{97} +14.5616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} - 3q^{5} + 3q^{7} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} - 3q^{5} + 3q^{7} + 9q^{8} + 7q^{10} - 4q^{11} + 10q^{14} + 3q^{16} - q^{17} - 6q^{19} + q^{20} - 2q^{22} - 4q^{23} + 3q^{25} + 16q^{28} - q^{29} - q^{31} + 9q^{32} - 9q^{34} + 4q^{35} - 11q^{37} + 14q^{38} - 5q^{40} + q^{41} + 5q^{43} - 10q^{44} - 2q^{46} - q^{49} - 24q^{50} - 11q^{53} + 6q^{55} + 22q^{56} + 25q^{58} - 14q^{59} + 16q^{61} + 8q^{62} + 7q^{64} - 5q^{67} - 11q^{68} + 2q^{70} + 28q^{71} + 12q^{73} + 3q^{74} + 2q^{76} - 6q^{77} + 15q^{79} + 21q^{80} + 9q^{82} - 10q^{83} - 7q^{85} - 6q^{86} - 18q^{88} + 18q^{89} - 10q^{92} - 34q^{94} + 26q^{95} + 13q^{97} + 25q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 0 0
\(4\) 4.56155 2.28078
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) 3.56155 1.34614 0.673070 0.739579i \(-0.264975\pi\)
0.673070 + 0.739579i \(0.264975\pi\)
\(8\) 6.56155 2.31986
\(9\) 0 0
\(10\) 1.43845 0.454877
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 9.12311 2.43825
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) −2.56155 −0.621268 −0.310634 0.950530i \(-0.600541\pi\)
−0.310634 + 0.950530i \(0.600541\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 2.56155 0.572781
\(21\) 0 0
\(22\) −5.12311 −1.09225
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 0 0
\(28\) 16.2462 3.07025
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 0 0
\(31\) 1.56155 0.280463 0.140232 0.990119i \(-0.455215\pi\)
0.140232 + 0.990119i \(0.455215\pi\)
\(32\) 6.56155 1.15993
\(33\) 0 0
\(34\) −6.56155 −1.12530
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −3.43845 −0.565277 −0.282639 0.959226i \(-0.591210\pi\)
−0.282639 + 0.959226i \(0.591210\pi\)
\(38\) 2.87689 0.466694
\(39\) 0 0
\(40\) 3.68466 0.582596
\(41\) 2.56155 0.400047 0.200024 0.979791i \(-0.435898\pi\)
0.200024 + 0.979791i \(0.435898\pi\)
\(42\) 0 0
\(43\) 0.438447 0.0668626 0.0334313 0.999441i \(-0.489357\pi\)
0.0334313 + 0.999441i \(0.489357\pi\)
\(44\) −9.12311 −1.37536
\(45\) 0 0
\(46\) −5.12311 −0.755361
\(47\) −8.24621 −1.20283 −0.601417 0.798935i \(-0.705397\pi\)
−0.601417 + 0.798935i \(0.705397\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) −12.0000 −1.69706
\(51\) 0 0
\(52\) 0 0
\(53\) −11.6847 −1.60501 −0.802506 0.596645i \(-0.796500\pi\)
−0.802506 + 0.596645i \(0.796500\pi\)
\(54\) 0 0
\(55\) −1.12311 −0.151440
\(56\) 23.3693 3.12286
\(57\) 0 0
\(58\) 14.5616 1.91203
\(59\) −11.1231 −1.44811 −0.724053 0.689745i \(-0.757723\pi\)
−0.724053 + 0.689745i \(0.757723\pi\)
\(60\) 0 0
\(61\) 12.1231 1.55220 0.776102 0.630607i \(-0.217194\pi\)
0.776102 + 0.630607i \(0.217194\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) 0 0
\(66\) 0 0
\(67\) −0.438447 −0.0535648 −0.0267824 0.999641i \(-0.508526\pi\)
−0.0267824 + 0.999641i \(0.508526\pi\)
\(68\) −11.6847 −1.41697
\(69\) 0 0
\(70\) 5.12311 0.612328
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 1.87689 0.219674 0.109837 0.993950i \(-0.464967\pi\)
0.109837 + 0.993950i \(0.464967\pi\)
\(74\) −8.80776 −1.02388
\(75\) 0 0
\(76\) 5.12311 0.587661
\(77\) −7.12311 −0.811753
\(78\) 0 0
\(79\) 9.56155 1.07576 0.537879 0.843022i \(-0.319226\pi\)
0.537879 + 0.843022i \(0.319226\pi\)
\(80\) 4.31534 0.482470
\(81\) 0 0
\(82\) 6.56155 0.724602
\(83\) −9.12311 −1.00139 −0.500695 0.865624i \(-0.666922\pi\)
−0.500695 + 0.865624i \(0.666922\pi\)
\(84\) 0 0
\(85\) −1.43845 −0.156022
\(86\) 1.12311 0.121108
\(87\) 0 0
\(88\) −13.1231 −1.39893
\(89\) 13.1231 1.39105 0.695523 0.718504i \(-0.255173\pi\)
0.695523 + 0.718504i \(0.255173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −9.12311 −0.951150
\(93\) 0 0
\(94\) −21.1231 −2.17868
\(95\) 0.630683 0.0647067
\(96\) 0 0
\(97\) 4.43845 0.450656 0.225328 0.974283i \(-0.427655\pi\)
0.225328 + 0.974283i \(0.427655\pi\)
\(98\) 14.5616 1.47094
\(99\) 0 0
\(100\) −21.3693 −2.13693
\(101\) 3.43845 0.342138 0.171069 0.985259i \(-0.445278\pi\)
0.171069 + 0.985259i \(0.445278\pi\)
\(102\) 0 0
\(103\) −7.56155 −0.745062 −0.372531 0.928020i \(-0.621510\pi\)
−0.372531 + 0.928020i \(0.621510\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −29.9309 −2.90714
\(107\) −8.24621 −0.797191 −0.398596 0.917127i \(-0.630502\pi\)
−0.398596 + 0.917127i \(0.630502\pi\)
\(108\) 0 0
\(109\) −17.8078 −1.70567 −0.852837 0.522177i \(-0.825120\pi\)
−0.852837 + 0.522177i \(0.825120\pi\)
\(110\) −2.87689 −0.274301
\(111\) 0 0
\(112\) 27.3693 2.58616
\(113\) 14.8078 1.39300 0.696499 0.717558i \(-0.254740\pi\)
0.696499 + 0.717558i \(0.254740\pi\)
\(114\) 0 0
\(115\) −1.12311 −0.104730
\(116\) 25.9309 2.40762
\(117\) 0 0
\(118\) −28.4924 −2.62294
\(119\) −9.12311 −0.836314
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 31.0540 2.81149
\(123\) 0 0
\(124\) 7.12311 0.639674
\(125\) −5.43845 −0.486430
\(126\) 0 0
\(127\) 9.56155 0.848451 0.424225 0.905557i \(-0.360546\pi\)
0.424225 + 0.905557i \(0.360546\pi\)
\(128\) −9.43845 −0.834249
\(129\) 0 0
\(130\) 0 0
\(131\) 17.3693 1.51756 0.758782 0.651345i \(-0.225795\pi\)
0.758782 + 0.651345i \(0.225795\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −1.12311 −0.0970215
\(135\) 0 0
\(136\) −16.8078 −1.44125
\(137\) −1.43845 −0.122895 −0.0614474 0.998110i \(-0.519572\pi\)
−0.0614474 + 0.998110i \(0.519572\pi\)
\(138\) 0 0
\(139\) 10.9309 0.927144 0.463572 0.886059i \(-0.346567\pi\)
0.463572 + 0.886059i \(0.346567\pi\)
\(140\) 9.12311 0.771043
\(141\) 0 0
\(142\) 35.8617 3.00945
\(143\) 0 0
\(144\) 0 0
\(145\) 3.19224 0.265101
\(146\) 4.80776 0.397893
\(147\) 0 0
\(148\) −15.6847 −1.28927
\(149\) −6.56155 −0.537543 −0.268772 0.963204i \(-0.586618\pi\)
−0.268772 + 0.963204i \(0.586618\pi\)
\(150\) 0 0
\(151\) −15.3693 −1.25074 −0.625369 0.780329i \(-0.715051\pi\)
−0.625369 + 0.780329i \(0.715051\pi\)
\(152\) 7.36932 0.597731
\(153\) 0 0
\(154\) −18.2462 −1.47032
\(155\) 0.876894 0.0704339
\(156\) 0 0
\(157\) −4.36932 −0.348709 −0.174355 0.984683i \(-0.555784\pi\)
−0.174355 + 0.984683i \(0.555784\pi\)
\(158\) 24.4924 1.94851
\(159\) 0 0
\(160\) 3.68466 0.291298
\(161\) −7.12311 −0.561379
\(162\) 0 0
\(163\) −15.8078 −1.23816 −0.619080 0.785328i \(-0.712494\pi\)
−0.619080 + 0.785328i \(0.712494\pi\)
\(164\) 11.6847 0.912419
\(165\) 0 0
\(166\) −23.3693 −1.81381
\(167\) −6.24621 −0.483346 −0.241673 0.970358i \(-0.577696\pi\)
−0.241673 + 0.970358i \(0.577696\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −3.68466 −0.282600
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −3.75379 −0.285395 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(174\) 0 0
\(175\) −16.6847 −1.26124
\(176\) −15.3693 −1.15851
\(177\) 0 0
\(178\) 33.6155 2.51959
\(179\) 13.1231 0.980867 0.490433 0.871479i \(-0.336838\pi\)
0.490433 + 0.871479i \(0.336838\pi\)
\(180\) 0 0
\(181\) 9.68466 0.719855 0.359927 0.932980i \(-0.382801\pi\)
0.359927 + 0.932980i \(0.382801\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −13.1231 −0.967448
\(185\) −1.93087 −0.141960
\(186\) 0 0
\(187\) 5.12311 0.374639
\(188\) −37.6155 −2.74339
\(189\) 0 0
\(190\) 1.61553 0.117203
\(191\) 0.876894 0.0634499 0.0317249 0.999497i \(-0.489900\pi\)
0.0317249 + 0.999497i \(0.489900\pi\)
\(192\) 0 0
\(193\) −19.4924 −1.40310 −0.701548 0.712623i \(-0.747507\pi\)
−0.701548 + 0.712623i \(0.747507\pi\)
\(194\) 11.3693 0.816269
\(195\) 0 0
\(196\) 25.9309 1.85220
\(197\) −11.3693 −0.810030 −0.405015 0.914310i \(-0.632734\pi\)
−0.405015 + 0.914310i \(0.632734\pi\)
\(198\) 0 0
\(199\) −23.1771 −1.64298 −0.821490 0.570223i \(-0.806857\pi\)
−0.821490 + 0.570223i \(0.806857\pi\)
\(200\) −30.7386 −2.17355
\(201\) 0 0
\(202\) 8.80776 0.619712
\(203\) 20.2462 1.42101
\(204\) 0 0
\(205\) 1.43845 0.100466
\(206\) −19.3693 −1.34952
\(207\) 0 0
\(208\) 0 0
\(209\) −2.24621 −0.155374
\(210\) 0 0
\(211\) 7.31534 0.503609 0.251804 0.967778i \(-0.418976\pi\)
0.251804 + 0.967778i \(0.418976\pi\)
\(212\) −53.3002 −3.66067
\(213\) 0 0
\(214\) −21.1231 −1.44395
\(215\) 0.246211 0.0167915
\(216\) 0 0
\(217\) 5.56155 0.377543
\(218\) −45.6155 −3.08947
\(219\) 0 0
\(220\) −5.12311 −0.345400
\(221\) 0 0
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 23.3693 1.56143
\(225\) 0 0
\(226\) 37.9309 2.52312
\(227\) −1.12311 −0.0745431 −0.0372716 0.999305i \(-0.511867\pi\)
−0.0372716 + 0.999305i \(0.511867\pi\)
\(228\) 0 0
\(229\) −0.246211 −0.0162701 −0.00813505 0.999967i \(-0.502589\pi\)
−0.00813505 + 0.999967i \(0.502589\pi\)
\(230\) −2.87689 −0.189697
\(231\) 0 0
\(232\) 37.3002 2.44888
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −4.63068 −0.302072
\(236\) −50.7386 −3.30280
\(237\) 0 0
\(238\) −23.3693 −1.51481
\(239\) −0.630683 −0.0407955 −0.0203977 0.999792i \(-0.506493\pi\)
−0.0203977 + 0.999792i \(0.506493\pi\)
\(240\) 0 0
\(241\) −2.80776 −0.180864 −0.0904320 0.995903i \(-0.528825\pi\)
−0.0904320 + 0.995903i \(0.528825\pi\)
\(242\) −17.9309 −1.15264
\(243\) 0 0
\(244\) 55.3002 3.54023
\(245\) 3.19224 0.203944
\(246\) 0 0
\(247\) 0 0
\(248\) 10.2462 0.650635
\(249\) 0 0
\(250\) −13.9309 −0.881066
\(251\) 30.7386 1.94021 0.970103 0.242695i \(-0.0780314\pi\)
0.970103 + 0.242695i \(0.0780314\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 24.4924 1.53679
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) 16.1771 1.00910 0.504549 0.863383i \(-0.331659\pi\)
0.504549 + 0.863383i \(0.331659\pi\)
\(258\) 0 0
\(259\) −12.2462 −0.760943
\(260\) 0 0
\(261\) 0 0
\(262\) 44.4924 2.74875
\(263\) 15.3693 0.947713 0.473856 0.880602i \(-0.342862\pi\)
0.473856 + 0.880602i \(0.342862\pi\)
\(264\) 0 0
\(265\) −6.56155 −0.403073
\(266\) 10.2462 0.628236
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) −3.36932 −0.205431 −0.102715 0.994711i \(-0.532753\pi\)
−0.102715 + 0.994711i \(0.532753\pi\)
\(270\) 0 0
\(271\) 1.06913 0.0649450 0.0324725 0.999473i \(-0.489662\pi\)
0.0324725 + 0.999473i \(0.489662\pi\)
\(272\) −19.6847 −1.19356
\(273\) 0 0
\(274\) −3.68466 −0.222598
\(275\) 9.36932 0.564991
\(276\) 0 0
\(277\) 17.6847 1.06257 0.531284 0.847194i \(-0.321710\pi\)
0.531284 + 0.847194i \(0.321710\pi\)
\(278\) 28.0000 1.67933
\(279\) 0 0
\(280\) 13.1231 0.784256
\(281\) −2.80776 −0.167497 −0.0837486 0.996487i \(-0.526689\pi\)
−0.0837486 + 0.996487i \(0.526689\pi\)
\(282\) 0 0
\(283\) −1.31534 −0.0781889 −0.0390945 0.999236i \(-0.512447\pi\)
−0.0390945 + 0.999236i \(0.512447\pi\)
\(284\) 63.8617 3.78950
\(285\) 0 0
\(286\) 0 0
\(287\) 9.12311 0.538520
\(288\) 0 0
\(289\) −10.4384 −0.614026
\(290\) 8.17708 0.480175
\(291\) 0 0
\(292\) 8.56155 0.501027
\(293\) 24.5616 1.43490 0.717451 0.696609i \(-0.245309\pi\)
0.717451 + 0.696609i \(0.245309\pi\)
\(294\) 0 0
\(295\) −6.24621 −0.363668
\(296\) −22.5616 −1.31136
\(297\) 0 0
\(298\) −16.8078 −0.973648
\(299\) 0 0
\(300\) 0 0
\(301\) 1.56155 0.0900064
\(302\) −39.3693 −2.26545
\(303\) 0 0
\(304\) 8.63068 0.495004
\(305\) 6.80776 0.389811
\(306\) 0 0
\(307\) −10.1922 −0.581702 −0.290851 0.956768i \(-0.593938\pi\)
−0.290851 + 0.956768i \(0.593938\pi\)
\(308\) −32.4924 −1.85143
\(309\) 0 0
\(310\) 2.24621 0.127576
\(311\) 10.8769 0.616772 0.308386 0.951261i \(-0.400211\pi\)
0.308386 + 0.951261i \(0.400211\pi\)
\(312\) 0 0
\(313\) −1.31534 −0.0743475 −0.0371738 0.999309i \(-0.511835\pi\)
−0.0371738 + 0.999309i \(0.511835\pi\)
\(314\) −11.1922 −0.631614
\(315\) 0 0
\(316\) 43.6155 2.45357
\(317\) −23.0540 −1.29484 −0.647420 0.762133i \(-0.724152\pi\)
−0.647420 + 0.762133i \(0.724152\pi\)
\(318\) 0 0
\(319\) −11.3693 −0.636560
\(320\) 0.807764 0.0451554
\(321\) 0 0
\(322\) −18.2462 −1.01682
\(323\) −2.87689 −0.160075
\(324\) 0 0
\(325\) 0 0
\(326\) −40.4924 −2.24267
\(327\) 0 0
\(328\) 16.8078 0.928054
\(329\) −29.3693 −1.61918
\(330\) 0 0
\(331\) 23.8078 1.30859 0.654297 0.756238i \(-0.272965\pi\)
0.654297 + 0.756238i \(0.272965\pi\)
\(332\) −41.6155 −2.28395
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) −0.246211 −0.0134520
\(336\) 0 0
\(337\) 2.12311 0.115653 0.0578265 0.998327i \(-0.481583\pi\)
0.0578265 + 0.998327i \(0.481583\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6.56155 −0.355850
\(341\) −3.12311 −0.169126
\(342\) 0 0
\(343\) −4.68466 −0.252948
\(344\) 2.87689 0.155112
\(345\) 0 0
\(346\) −9.61553 −0.516934
\(347\) 13.6155 0.730920 0.365460 0.930827i \(-0.380912\pi\)
0.365460 + 0.930827i \(0.380912\pi\)
\(348\) 0 0
\(349\) 13.8078 0.739113 0.369556 0.929208i \(-0.379510\pi\)
0.369556 + 0.929208i \(0.379510\pi\)
\(350\) −42.7386 −2.28448
\(351\) 0 0
\(352\) −13.1231 −0.699464
\(353\) 17.6847 0.941259 0.470630 0.882331i \(-0.344027\pi\)
0.470630 + 0.882331i \(0.344027\pi\)
\(354\) 0 0
\(355\) 7.86174 0.417258
\(356\) 59.8617 3.17267
\(357\) 0 0
\(358\) 33.6155 1.77664
\(359\) 15.3693 0.811162 0.405581 0.914059i \(-0.367069\pi\)
0.405581 + 0.914059i \(0.367069\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 24.8078 1.30387
\(363\) 0 0
\(364\) 0 0
\(365\) 1.05398 0.0551676
\(366\) 0 0
\(367\) 20.0540 1.04681 0.523404 0.852084i \(-0.324662\pi\)
0.523404 + 0.852084i \(0.324662\pi\)
\(368\) −15.3693 −0.801181
\(369\) 0 0
\(370\) −4.94602 −0.257132
\(371\) −41.6155 −2.16057
\(372\) 0 0
\(373\) 3.63068 0.187990 0.0939948 0.995573i \(-0.470036\pi\)
0.0939948 + 0.995573i \(0.470036\pi\)
\(374\) 13.1231 0.678580
\(375\) 0 0
\(376\) −54.1080 −2.79040
\(377\) 0 0
\(378\) 0 0
\(379\) −11.3153 −0.581230 −0.290615 0.956840i \(-0.593860\pi\)
−0.290615 + 0.956840i \(0.593860\pi\)
\(380\) 2.87689 0.147582
\(381\) 0 0
\(382\) 2.24621 0.114926
\(383\) 26.7386 1.36628 0.683140 0.730287i \(-0.260614\pi\)
0.683140 + 0.730287i \(0.260614\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) −49.9309 −2.54141
\(387\) 0 0
\(388\) 20.2462 1.02785
\(389\) 3.05398 0.154843 0.0774213 0.996998i \(-0.475331\pi\)
0.0774213 + 0.996998i \(0.475331\pi\)
\(390\) 0 0
\(391\) 5.12311 0.259087
\(392\) 37.3002 1.88394
\(393\) 0 0
\(394\) −29.1231 −1.46720
\(395\) 5.36932 0.270160
\(396\) 0 0
\(397\) 12.0540 0.604972 0.302486 0.953154i \(-0.402184\pi\)
0.302486 + 0.953154i \(0.402184\pi\)
\(398\) −59.3693 −2.97591
\(399\) 0 0
\(400\) −36.0000 −1.80000
\(401\) 18.5616 0.926920 0.463460 0.886118i \(-0.346608\pi\)
0.463460 + 0.886118i \(0.346608\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 15.6847 0.780341
\(405\) 0 0
\(406\) 51.8617 2.57385
\(407\) 6.87689 0.340875
\(408\) 0 0
\(409\) 18.3693 0.908304 0.454152 0.890924i \(-0.349942\pi\)
0.454152 + 0.890924i \(0.349942\pi\)
\(410\) 3.68466 0.181972
\(411\) 0 0
\(412\) −34.4924 −1.69932
\(413\) −39.6155 −1.94935
\(414\) 0 0
\(415\) −5.12311 −0.251483
\(416\) 0 0
\(417\) 0 0
\(418\) −5.75379 −0.281427
\(419\) 17.7538 0.867329 0.433665 0.901074i \(-0.357220\pi\)
0.433665 + 0.901074i \(0.357220\pi\)
\(420\) 0 0
\(421\) −14.7538 −0.719056 −0.359528 0.933134i \(-0.617062\pi\)
−0.359528 + 0.933134i \(0.617062\pi\)
\(422\) 18.7386 0.912182
\(423\) 0 0
\(424\) −76.6695 −3.72340
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 43.1771 2.08949
\(428\) −37.6155 −1.81822
\(429\) 0 0
\(430\) 0.630683 0.0304142
\(431\) −2.87689 −0.138575 −0.0692876 0.997597i \(-0.522073\pi\)
−0.0692876 + 0.997597i \(0.522073\pi\)
\(432\) 0 0
\(433\) 25.2462 1.21326 0.606628 0.794986i \(-0.292522\pi\)
0.606628 + 0.794986i \(0.292522\pi\)
\(434\) 14.2462 0.683840
\(435\) 0 0
\(436\) −81.2311 −3.89026
\(437\) −2.24621 −0.107451
\(438\) 0 0
\(439\) −1.31534 −0.0627778 −0.0313889 0.999507i \(-0.509993\pi\)
−0.0313889 + 0.999507i \(0.509993\pi\)
\(440\) −7.36932 −0.351318
\(441\) 0 0
\(442\) 0 0
\(443\) −14.7386 −0.700254 −0.350127 0.936702i \(-0.613862\pi\)
−0.350127 + 0.936702i \(0.613862\pi\)
\(444\) 0 0
\(445\) 7.36932 0.349339
\(446\) −20.4924 −0.970344
\(447\) 0 0
\(448\) 5.12311 0.242044
\(449\) 8.24621 0.389163 0.194581 0.980886i \(-0.437665\pi\)
0.194581 + 0.980886i \(0.437665\pi\)
\(450\) 0 0
\(451\) −5.12311 −0.241238
\(452\) 67.5464 3.17712
\(453\) 0 0
\(454\) −2.87689 −0.135019
\(455\) 0 0
\(456\) 0 0
\(457\) 28.6155 1.33858 0.669289 0.743002i \(-0.266599\pi\)
0.669289 + 0.743002i \(0.266599\pi\)
\(458\) −0.630683 −0.0294699
\(459\) 0 0
\(460\) −5.12311 −0.238866
\(461\) −36.8078 −1.71431 −0.857154 0.515060i \(-0.827770\pi\)
−0.857154 + 0.515060i \(0.827770\pi\)
\(462\) 0 0
\(463\) 26.6847 1.24014 0.620071 0.784546i \(-0.287104\pi\)
0.620071 + 0.784546i \(0.287104\pi\)
\(464\) 43.6847 2.02801
\(465\) 0 0
\(466\) −66.6004 −3.08520
\(467\) 26.0000 1.20314 0.601568 0.798821i \(-0.294543\pi\)
0.601568 + 0.798821i \(0.294543\pi\)
\(468\) 0 0
\(469\) −1.56155 −0.0721058
\(470\) −11.8617 −0.547141
\(471\) 0 0
\(472\) −72.9848 −3.35940
\(473\) −0.876894 −0.0403196
\(474\) 0 0
\(475\) −5.26137 −0.241408
\(476\) −41.6155 −1.90744
\(477\) 0 0
\(478\) −1.61553 −0.0738925
\(479\) −6.24621 −0.285397 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −7.19224 −0.327597
\(483\) 0 0
\(484\) −31.9309 −1.45140
\(485\) 2.49242 0.113175
\(486\) 0 0
\(487\) 1.12311 0.0508928 0.0254464 0.999676i \(-0.491899\pi\)
0.0254464 + 0.999676i \(0.491899\pi\)
\(488\) 79.5464 3.60090
\(489\) 0 0
\(490\) 8.17708 0.369403
\(491\) 19.7538 0.891476 0.445738 0.895163i \(-0.352941\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(492\) 0 0
\(493\) −14.5616 −0.655819
\(494\) 0 0
\(495\) 0 0
\(496\) 12.0000 0.538816
\(497\) 49.8617 2.23660
\(498\) 0 0
\(499\) 28.4924 1.27550 0.637748 0.770245i \(-0.279866\pi\)
0.637748 + 0.770245i \(0.279866\pi\)
\(500\) −24.8078 −1.10944
\(501\) 0 0
\(502\) 78.7386 3.51428
\(503\) 11.7538 0.524076 0.262038 0.965058i \(-0.415605\pi\)
0.262038 + 0.965058i \(0.415605\pi\)
\(504\) 0 0
\(505\) 1.93087 0.0859226
\(506\) 10.2462 0.455500
\(507\) 0 0
\(508\) 43.6155 1.93513
\(509\) 6.80776 0.301749 0.150874 0.988553i \(-0.451791\pi\)
0.150874 + 0.988553i \(0.451791\pi\)
\(510\) 0 0
\(511\) 6.68466 0.295712
\(512\) −50.4233 −2.22842
\(513\) 0 0
\(514\) 41.4384 1.82777
\(515\) −4.24621 −0.187110
\(516\) 0 0
\(517\) 16.4924 0.725336
\(518\) −31.3693 −1.37829
\(519\) 0 0
\(520\) 0 0
\(521\) 37.9309 1.66178 0.830891 0.556436i \(-0.187831\pi\)
0.830891 + 0.556436i \(0.187831\pi\)
\(522\) 0 0
\(523\) −23.8617 −1.04340 −0.521701 0.853129i \(-0.674702\pi\)
−0.521701 + 0.853129i \(0.674702\pi\)
\(524\) 79.2311 3.46122
\(525\) 0 0
\(526\) 39.3693 1.71658
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −16.8078 −0.730083
\(531\) 0 0
\(532\) 18.2462 0.791074
\(533\) 0 0
\(534\) 0 0
\(535\) −4.63068 −0.200202
\(536\) −2.87689 −0.124263
\(537\) 0 0
\(538\) −8.63068 −0.372095
\(539\) −11.3693 −0.489711
\(540\) 0 0
\(541\) 29.7386 1.27856 0.639282 0.768972i \(-0.279232\pi\)
0.639282 + 0.768972i \(0.279232\pi\)
\(542\) 2.73863 0.117634
\(543\) 0 0
\(544\) −16.8078 −0.720627
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −24.9309 −1.06597 −0.532984 0.846126i \(-0.678929\pi\)
−0.532984 + 0.846126i \(0.678929\pi\)
\(548\) −6.56155 −0.280296
\(549\) 0 0
\(550\) 24.0000 1.02336
\(551\) 6.38447 0.271988
\(552\) 0 0
\(553\) 34.0540 1.44812
\(554\) 45.3002 1.92462
\(555\) 0 0
\(556\) 49.8617 2.11461
\(557\) −14.0691 −0.596128 −0.298064 0.954546i \(-0.596341\pi\)
−0.298064 + 0.954546i \(0.596341\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 15.3693 0.649472
\(561\) 0 0
\(562\) −7.19224 −0.303386
\(563\) −1.36932 −0.0577098 −0.0288549 0.999584i \(-0.509186\pi\)
−0.0288549 + 0.999584i \(0.509186\pi\)
\(564\) 0 0
\(565\) 8.31534 0.349829
\(566\) −3.36932 −0.141623
\(567\) 0 0
\(568\) 91.8617 3.85443
\(569\) −40.7386 −1.70785 −0.853926 0.520394i \(-0.825785\pi\)
−0.853926 + 0.520394i \(0.825785\pi\)
\(570\) 0 0
\(571\) 19.3693 0.810581 0.405290 0.914188i \(-0.367170\pi\)
0.405290 + 0.914188i \(0.367170\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 23.3693 0.975416
\(575\) 9.36932 0.390728
\(576\) 0 0
\(577\) 29.6847 1.23579 0.617894 0.786261i \(-0.287986\pi\)
0.617894 + 0.786261i \(0.287986\pi\)
\(578\) −26.7386 −1.11218
\(579\) 0 0
\(580\) 14.5616 0.604636
\(581\) −32.4924 −1.34801
\(582\) 0 0
\(583\) 23.3693 0.967858
\(584\) 12.3153 0.509612
\(585\) 0 0
\(586\) 62.9157 2.59902
\(587\) 14.6307 0.603873 0.301936 0.953328i \(-0.402367\pi\)
0.301936 + 0.953328i \(0.402367\pi\)
\(588\) 0 0
\(589\) 1.75379 0.0722636
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) −26.4233 −1.08599
\(593\) 44.4233 1.82425 0.912123 0.409917i \(-0.134442\pi\)
0.912123 + 0.409917i \(0.134442\pi\)
\(594\) 0 0
\(595\) −5.12311 −0.210027
\(596\) −29.9309 −1.22602
\(597\) 0 0
\(598\) 0 0
\(599\) 0.384472 0.0157091 0.00785455 0.999969i \(-0.497500\pi\)
0.00785455 + 0.999969i \(0.497500\pi\)
\(600\) 0 0
\(601\) −35.9309 −1.46565 −0.732825 0.680417i \(-0.761799\pi\)
−0.732825 + 0.680417i \(0.761799\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) −70.1080 −2.85265
\(605\) −3.93087 −0.159813
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 7.36932 0.298865
\(609\) 0 0
\(610\) 17.4384 0.706062
\(611\) 0 0
\(612\) 0 0
\(613\) −22.8617 −0.923377 −0.461688 0.887042i \(-0.652756\pi\)
−0.461688 + 0.887042i \(0.652756\pi\)
\(614\) −26.1080 −1.05363
\(615\) 0 0
\(616\) −46.7386 −1.88315
\(617\) −10.8078 −0.435104 −0.217552 0.976049i \(-0.569807\pi\)
−0.217552 + 0.976049i \(0.569807\pi\)
\(618\) 0 0
\(619\) 24.3002 0.976707 0.488353 0.872646i \(-0.337598\pi\)
0.488353 + 0.872646i \(0.337598\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 27.8617 1.11715
\(623\) 46.7386 1.87254
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −3.36932 −0.134665
\(627\) 0 0
\(628\) −19.9309 −0.795328
\(629\) 8.80776 0.351189
\(630\) 0 0
\(631\) 14.4384 0.574786 0.287393 0.957813i \(-0.407212\pi\)
0.287393 + 0.957813i \(0.407212\pi\)
\(632\) 62.7386 2.49561
\(633\) 0 0
\(634\) −59.0540 −2.34533
\(635\) 5.36932 0.213075
\(636\) 0 0
\(637\) 0 0
\(638\) −29.1231 −1.15299
\(639\) 0 0
\(640\) −5.30019 −0.209508
\(641\) −26.1771 −1.03393 −0.516966 0.856006i \(-0.672939\pi\)
−0.516966 + 0.856006i \(0.672939\pi\)
\(642\) 0 0
\(643\) −38.5464 −1.52012 −0.760061 0.649852i \(-0.774831\pi\)
−0.760061 + 0.649852i \(0.774831\pi\)
\(644\) −32.4924 −1.28038
\(645\) 0 0
\(646\) −7.36932 −0.289942
\(647\) 47.6155 1.87196 0.935980 0.352054i \(-0.114517\pi\)
0.935980 + 0.352054i \(0.114517\pi\)
\(648\) 0 0
\(649\) 22.2462 0.873240
\(650\) 0 0
\(651\) 0 0
\(652\) −72.1080 −2.82397
\(653\) −14.8769 −0.582178 −0.291089 0.956696i \(-0.594018\pi\)
−0.291089 + 0.956696i \(0.594018\pi\)
\(654\) 0 0
\(655\) 9.75379 0.381112
\(656\) 19.6847 0.768557
\(657\) 0 0
\(658\) −75.2311 −2.93281
\(659\) −14.2462 −0.554954 −0.277477 0.960732i \(-0.589498\pi\)
−0.277477 + 0.960732i \(0.589498\pi\)
\(660\) 0 0
\(661\) −30.3693 −1.18123 −0.590615 0.806954i \(-0.701115\pi\)
−0.590615 + 0.806954i \(0.701115\pi\)
\(662\) 60.9848 2.37024
\(663\) 0 0
\(664\) −59.8617 −2.32309
\(665\) 2.24621 0.0871043
\(666\) 0 0
\(667\) −11.3693 −0.440222
\(668\) −28.4924 −1.10240
\(669\) 0 0
\(670\) −0.630683 −0.0243654
\(671\) −24.2462 −0.936015
\(672\) 0 0
\(673\) −6.75379 −0.260339 −0.130170 0.991492i \(-0.541552\pi\)
−0.130170 + 0.991492i \(0.541552\pi\)
\(674\) 5.43845 0.209481
\(675\) 0 0
\(676\) 0 0
\(677\) −25.6155 −0.984485 −0.492242 0.870458i \(-0.663823\pi\)
−0.492242 + 0.870458i \(0.663823\pi\)
\(678\) 0 0
\(679\) 15.8078 0.606646
\(680\) −9.43845 −0.361948
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) 36.1080 1.38163 0.690816 0.723030i \(-0.257251\pi\)
0.690816 + 0.723030i \(0.257251\pi\)
\(684\) 0 0
\(685\) −0.807764 −0.0308631
\(686\) −12.0000 −0.458162
\(687\) 0 0
\(688\) 3.36932 0.128454
\(689\) 0 0
\(690\) 0 0
\(691\) −2.30019 −0.0875032 −0.0437516 0.999042i \(-0.513931\pi\)
−0.0437516 + 0.999042i \(0.513931\pi\)
\(692\) −17.1231 −0.650923
\(693\) 0 0
\(694\) 34.8769 1.32391
\(695\) 6.13826 0.232837
\(696\) 0 0
\(697\) −6.56155 −0.248537
\(698\) 35.3693 1.33875
\(699\) 0 0
\(700\) −76.1080 −2.87661
\(701\) −19.3693 −0.731569 −0.365785 0.930700i \(-0.619199\pi\)
−0.365785 + 0.930700i \(0.619199\pi\)
\(702\) 0 0
\(703\) −3.86174 −0.145648
\(704\) −2.87689 −0.108427
\(705\) 0 0
\(706\) 45.3002 1.70490
\(707\) 12.2462 0.460566
\(708\) 0 0
\(709\) −25.4924 −0.957388 −0.478694 0.877982i \(-0.658890\pi\)
−0.478694 + 0.877982i \(0.658890\pi\)
\(710\) 20.1383 0.755775
\(711\) 0 0
\(712\) 86.1080 3.22703
\(713\) −3.12311 −0.116961
\(714\) 0 0
\(715\) 0 0
\(716\) 59.8617 2.23714
\(717\) 0 0
\(718\) 39.3693 1.46925
\(719\) −1.36932 −0.0510669 −0.0255335 0.999674i \(-0.508128\pi\)
−0.0255335 + 0.999674i \(0.508128\pi\)
\(720\) 0 0
\(721\) −26.9309 −1.00296
\(722\) −45.4384 −1.69104
\(723\) 0 0
\(724\) 44.1771 1.64183
\(725\) −26.6307 −0.989039
\(726\) 0 0
\(727\) 39.6695 1.47126 0.735630 0.677383i \(-0.236886\pi\)
0.735630 + 0.677383i \(0.236886\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.69981 0.0999246
\(731\) −1.12311 −0.0415396
\(732\) 0 0
\(733\) −53.4924 −1.97579 −0.987894 0.155131i \(-0.950420\pi\)
−0.987894 + 0.155131i \(0.950420\pi\)
\(734\) 51.3693 1.89608
\(735\) 0 0
\(736\) −13.1231 −0.483724
\(737\) 0.876894 0.0323008
\(738\) 0 0
\(739\) 6.24621 0.229771 0.114885 0.993379i \(-0.463350\pi\)
0.114885 + 0.993379i \(0.463350\pi\)
\(740\) −8.80776 −0.323780
\(741\) 0 0
\(742\) −106.600 −3.91342
\(743\) −37.3693 −1.37095 −0.685474 0.728097i \(-0.740405\pi\)
−0.685474 + 0.728097i \(0.740405\pi\)
\(744\) 0 0
\(745\) −3.68466 −0.134995
\(746\) 9.30019 0.340504
\(747\) 0 0
\(748\) 23.3693 0.854467
\(749\) −29.3693 −1.07313
\(750\) 0 0
\(751\) −30.1080 −1.09865 −0.549327 0.835607i \(-0.685116\pi\)
−0.549327 + 0.835607i \(0.685116\pi\)
\(752\) −63.3693 −2.31084
\(753\) 0 0
\(754\) 0 0
\(755\) −8.63068 −0.314103
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) −28.9848 −1.05278
\(759\) 0 0
\(760\) 4.13826 0.150110
\(761\) −15.3693 −0.557137 −0.278569 0.960416i \(-0.589860\pi\)
−0.278569 + 0.960416i \(0.589860\pi\)
\(762\) 0 0
\(763\) −63.4233 −2.29608
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) 68.4924 2.47473
\(767\) 0 0
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) −10.2462 −0.369248
\(771\) 0 0
\(772\) −88.9157 −3.20015
\(773\) −7.75379 −0.278884 −0.139442 0.990230i \(-0.544531\pi\)
−0.139442 + 0.990230i \(0.544531\pi\)
\(774\) 0 0
\(775\) −7.31534 −0.262775
\(776\) 29.1231 1.04546
\(777\) 0 0
\(778\) 7.82292 0.280465
\(779\) 2.87689 0.103075
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) 13.1231 0.469281
\(783\) 0 0
\(784\) 43.6847 1.56017
\(785\) −2.45360 −0.0875728
\(786\) 0 0
\(787\) 1.17708 0.0419584 0.0209792 0.999780i \(-0.493322\pi\)
0.0209792 + 0.999780i \(0.493322\pi\)
\(788\) −51.8617 −1.84750
\(789\) 0 0
\(790\) 13.7538 0.489338
\(791\) 52.7386 1.87517
\(792\) 0 0
\(793\) 0 0
\(794\) 30.8769 1.09578
\(795\) 0 0
\(796\) −105.723 −3.74727
\(797\) 41.6155 1.47410 0.737049 0.675840i \(-0.236219\pi\)
0.737049 + 0.675840i \(0.236219\pi\)
\(798\) 0 0
\(799\) 21.1231 0.747282
\(800\) −30.7386 −1.08677
\(801\) 0 0
\(802\) 47.5464 1.67892
\(803\) −3.75379 −0.132468
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 0 0
\(808\) 22.5616 0.793713
\(809\) −37.3002 −1.31140 −0.655702 0.755019i \(-0.727627\pi\)
−0.655702 + 0.755019i \(0.727627\pi\)
\(810\) 0 0
\(811\) 1.56155 0.0548335 0.0274168 0.999624i \(-0.491272\pi\)
0.0274168 + 0.999624i \(0.491272\pi\)
\(812\) 92.3542 3.24100
\(813\) 0 0
\(814\) 17.6155 0.617424
\(815\) −8.87689 −0.310944
\(816\) 0 0
\(817\) 0.492423 0.0172277
\(818\) 47.0540 1.64520
\(819\) 0 0
\(820\) 6.56155 0.229139
\(821\) −26.4924 −0.924592 −0.462296 0.886726i \(-0.652974\pi\)
−0.462296 + 0.886726i \(0.652974\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −49.6155 −1.72844
\(825\) 0 0
\(826\) −101.477 −3.53085
\(827\) 34.7386 1.20798 0.603990 0.796992i \(-0.293577\pi\)
0.603990 + 0.796992i \(0.293577\pi\)
\(828\) 0 0
\(829\) −19.4924 −0.677000 −0.338500 0.940966i \(-0.609919\pi\)
−0.338500 + 0.940966i \(0.609919\pi\)
\(830\) −13.1231 −0.455510
\(831\) 0 0
\(832\) 0 0
\(833\) −14.5616 −0.504528
\(834\) 0 0
\(835\) −3.50758 −0.121385
\(836\) −10.2462 −0.354373
\(837\) 0 0
\(838\) 45.4773 1.57099
\(839\) −19.6155 −0.677203 −0.338602 0.940930i \(-0.609954\pi\)
−0.338602 + 0.940930i \(0.609954\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) −37.7926 −1.30242
\(843\) 0 0
\(844\) 33.3693 1.14862
\(845\) 0 0
\(846\) 0 0
\(847\) −24.9309 −0.856635
\(848\) −89.7926 −3.08349
\(849\) 0 0
\(850\) 30.7386 1.05433
\(851\) 6.87689 0.235737
\(852\) 0 0
\(853\) 6.12311 0.209651 0.104826 0.994491i \(-0.466572\pi\)
0.104826 + 0.994491i \(0.466572\pi\)
\(854\) 110.600 3.78467
\(855\) 0 0
\(856\) −54.1080 −1.84937
\(857\) −31.4384 −1.07392 −0.536958 0.843609i \(-0.680427\pi\)
−0.536958 + 0.843609i \(0.680427\pi\)
\(858\) 0 0
\(859\) 20.4384 0.697351 0.348675 0.937244i \(-0.386632\pi\)
0.348675 + 0.937244i \(0.386632\pi\)
\(860\) 1.12311 0.0382976
\(861\) 0 0
\(862\) −7.36932 −0.251000
\(863\) −2.49242 −0.0848430 −0.0424215 0.999100i \(-0.513507\pi\)
−0.0424215 + 0.999100i \(0.513507\pi\)
\(864\) 0 0
\(865\) −2.10795 −0.0716725
\(866\) 64.6695 2.19756
\(867\) 0 0
\(868\) 25.3693 0.861091
\(869\) −19.1231 −0.648707
\(870\) 0 0
\(871\) 0 0
\(872\) −116.847 −3.95692
\(873\) 0 0
\(874\) −5.75379 −0.194625
\(875\) −19.3693 −0.654802
\(876\) 0 0
\(877\) −19.4384 −0.656390 −0.328195 0.944610i \(-0.606440\pi\)
−0.328195 + 0.944610i \(0.606440\pi\)
\(878\) −3.36932 −0.113709
\(879\) 0 0
\(880\) −8.63068 −0.290940
\(881\) 37.9309 1.27792 0.638962 0.769239i \(-0.279364\pi\)
0.638962 + 0.769239i \(0.279364\pi\)
\(882\) 0 0
\(883\) −11.8078 −0.397363 −0.198681 0.980064i \(-0.563666\pi\)
−0.198681 + 0.980064i \(0.563666\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −37.7538 −1.26836
\(887\) −49.3693 −1.65766 −0.828830 0.559501i \(-0.810993\pi\)
−0.828830 + 0.559501i \(0.810993\pi\)
\(888\) 0 0
\(889\) 34.0540 1.14213
\(890\) 18.8769 0.632755
\(891\) 0 0
\(892\) −36.4924 −1.22186
\(893\) −9.26137 −0.309920
\(894\) 0 0
\(895\) 7.36932 0.246329
\(896\) −33.6155 −1.12302
\(897\) 0 0
\(898\) 21.1231 0.704887
\(899\) 8.87689 0.296061
\(900\) 0 0
\(901\) 29.9309 0.997142
\(902\) −13.1231 −0.436952
\(903\) 0 0
\(904\) 97.1619 3.23156
\(905\) 5.43845 0.180780
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −5.12311 −0.170016
\(909\) 0 0
\(910\) 0 0
\(911\) 10.7386 0.355787 0.177893 0.984050i \(-0.443072\pi\)
0.177893 + 0.984050i \(0.443072\pi\)
\(912\) 0 0
\(913\) 18.2462 0.603861
\(914\) 73.3002 2.42455
\(915\) 0 0
\(916\) −1.12311 −0.0371085
\(917\) 61.8617 2.04285
\(918\) 0 0
\(919\) 44.4924 1.46767 0.733835 0.679328i \(-0.237729\pi\)
0.733835 + 0.679328i \(0.237729\pi\)
\(920\) −7.36932 −0.242959
\(921\) 0 0
\(922\) −94.2850 −3.10511
\(923\) 0 0
\(924\) 0 0
\(925\) 16.1080 0.529626
\(926\) 68.3542 2.24626
\(927\) 0 0
\(928\) 37.3002 1.22444
\(929\) 12.8078 0.420209 0.210105 0.977679i \(-0.432620\pi\)
0.210105 + 0.977679i \(0.432620\pi\)
\(930\) 0 0
\(931\) 6.38447 0.209243
\(932\) −118.600 −3.88488
\(933\) 0 0
\(934\) 66.6004 2.17923
\(935\) 2.87689 0.0940845
\(936\) 0 0
\(937\) −3.43845 −0.112329 −0.0561646 0.998422i \(-0.517887\pi\)
−0.0561646 + 0.998422i \(0.517887\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −21.1231 −0.688960
\(941\) −2.49242 −0.0812507 −0.0406253 0.999174i \(-0.512935\pi\)
−0.0406253 + 0.999174i \(0.512935\pi\)
\(942\) 0 0
\(943\) −5.12311 −0.166831
\(944\) −85.4773 −2.78205
\(945\) 0 0
\(946\) −2.24621 −0.0730306
\(947\) 10.7386 0.348959 0.174479 0.984661i \(-0.444176\pi\)
0.174479 + 0.984661i \(0.444176\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −13.4773 −0.437260
\(951\) 0 0
\(952\) −59.8617 −1.94013
\(953\) 34.9848 1.13327 0.566635 0.823969i \(-0.308245\pi\)
0.566635 + 0.823969i \(0.308245\pi\)
\(954\) 0 0
\(955\) 0.492423 0.0159344
\(956\) −2.87689 −0.0930454
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) −5.12311 −0.165434
\(960\) 0 0
\(961\) −28.5616 −0.921340
\(962\) 0 0
\(963\) 0 0
\(964\) −12.8078 −0.412510
\(965\) −10.9460 −0.352365
\(966\) 0 0
\(967\) 9.12311 0.293379 0.146690 0.989183i \(-0.453138\pi\)
0.146690 + 0.989183i \(0.453138\pi\)
\(968\) −45.9309 −1.47627
\(969\) 0 0
\(970\) 6.38447 0.204993
\(971\) −52.9848 −1.70036 −0.850182 0.526488i \(-0.823508\pi\)
−0.850182 + 0.526488i \(0.823508\pi\)
\(972\) 0 0
\(973\) 38.9309 1.24807
\(974\) 2.87689 0.0921816
\(975\) 0 0
\(976\) 93.1619 2.98204
\(977\) 15.8229 0.506220 0.253110 0.967438i \(-0.418547\pi\)
0.253110 + 0.967438i \(0.418547\pi\)
\(978\) 0 0
\(979\) −26.2462 −0.838833
\(980\) 14.5616 0.465152
\(981\) 0 0
\(982\) 50.6004 1.61472
\(983\) −27.6155 −0.880799 −0.440399 0.897802i \(-0.645163\pi\)
−0.440399 + 0.897802i \(0.645163\pi\)
\(984\) 0 0
\(985\) −6.38447 −0.203426
\(986\) −37.3002 −1.18788
\(987\) 0 0
\(988\) 0 0
\(989\) −0.876894 −0.0278836
\(990\) 0 0
\(991\) 40.3542 1.28189 0.640946 0.767586i \(-0.278542\pi\)
0.640946 + 0.767586i \(0.278542\pi\)
\(992\) 10.2462 0.325318
\(993\) 0 0
\(994\) 127.723 4.05114
\(995\) −13.0152 −0.412608
\(996\) 0 0
\(997\) 20.6155 0.652900 0.326450 0.945214i \(-0.394147\pi\)
0.326450 + 0.945214i \(0.394147\pi\)
\(998\) 72.9848 2.31029
\(999\) 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.m.1.2 2
3.2 odd 2 507.2.a.d.1.1 2
12.11 even 2 8112.2.a.bo.1.1 2
13.4 even 6 117.2.g.c.55.2 4
13.5 odd 4 1521.2.b.h.1351.1 4
13.8 odd 4 1521.2.b.h.1351.4 4
13.10 even 6 117.2.g.c.100.2 4
13.12 even 2 1521.2.a.g.1.1 2
39.2 even 12 507.2.j.g.316.1 8
39.5 even 4 507.2.b.d.337.4 4
39.8 even 4 507.2.b.d.337.1 4
39.11 even 12 507.2.j.g.316.4 8
39.17 odd 6 39.2.e.b.16.1 4
39.20 even 12 507.2.j.g.361.1 8
39.23 odd 6 39.2.e.b.22.1 yes 4
39.29 odd 6 507.2.e.g.22.2 4
39.32 even 12 507.2.j.g.361.4 8
39.35 odd 6 507.2.e.g.484.2 4
39.38 odd 2 507.2.a.g.1.2 2
52.23 odd 6 1872.2.t.r.1153.1 4
52.43 odd 6 1872.2.t.r.289.1 4
156.23 even 6 624.2.q.h.529.2 4
156.95 even 6 624.2.q.h.289.2 4
156.155 even 2 8112.2.a.bk.1.2 2
195.17 even 12 975.2.bb.i.874.4 8
195.23 even 12 975.2.bb.i.724.4 8
195.62 even 12 975.2.bb.i.724.1 8
195.134 odd 6 975.2.i.k.601.2 4
195.173 even 12 975.2.bb.i.874.1 8
195.179 odd 6 975.2.i.k.451.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.b.16.1 4 39.17 odd 6
39.2.e.b.22.1 yes 4 39.23 odd 6
117.2.g.c.55.2 4 13.4 even 6
117.2.g.c.100.2 4 13.10 even 6
507.2.a.d.1.1 2 3.2 odd 2
507.2.a.g.1.2 2 39.38 odd 2
507.2.b.d.337.1 4 39.8 even 4
507.2.b.d.337.4 4 39.5 even 4
507.2.e.g.22.2 4 39.29 odd 6
507.2.e.g.484.2 4 39.35 odd 6
507.2.j.g.316.1 8 39.2 even 12
507.2.j.g.316.4 8 39.11 even 12
507.2.j.g.361.1 8 39.20 even 12
507.2.j.g.361.4 8 39.32 even 12
624.2.q.h.289.2 4 156.95 even 6
624.2.q.h.529.2 4 156.23 even 6
975.2.i.k.451.2 4 195.179 odd 6
975.2.i.k.601.2 4 195.134 odd 6
975.2.bb.i.724.1 8 195.62 even 12
975.2.bb.i.724.4 8 195.23 even 12
975.2.bb.i.874.1 8 195.173 even 12
975.2.bb.i.874.4 8 195.17 even 12
1521.2.a.g.1.1 2 13.12 even 2
1521.2.a.m.1.2 2 1.1 even 1 trivial
1521.2.b.h.1351.1 4 13.5 odd 4
1521.2.b.h.1351.4 4 13.8 odd 4
1872.2.t.r.289.1 4 52.43 odd 6
1872.2.t.r.1153.1 4 52.23 odd 6
8112.2.a.bk.1.2 2 156.155 even 2
8112.2.a.bo.1.1 2 12.11 even 2