Properties

Label 4.18.a.a.1.2
Level 4
Weight 18
Character 4.1
Self dual yes
Analytic conductor 7.329
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.32888349378\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{9361}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-47.8761\)
Character \(\chi\) = 4.1

$q$-expansion

\(f(q)\) \(=\) \(q+15636.4 q^{3} -366730. q^{5} +2.37095e7 q^{7} +1.15358e8 q^{9} +O(q^{10})\) \(q+15636.4 q^{3} -366730. q^{5} +2.37095e7 q^{7} +1.15358e8 q^{9} +1.33786e9 q^{11} -3.68345e9 q^{13} -5.73434e9 q^{15} -1.70006e10 q^{17} -2.59141e9 q^{19} +3.70732e11 q^{21} -1.96178e11 q^{23} -6.28449e11 q^{25} -2.15505e11 q^{27} -1.36209e12 q^{29} -2.25575e12 q^{31} +2.09194e13 q^{33} -8.69497e12 q^{35} +1.28643e12 q^{37} -5.75961e13 q^{39} +3.74268e13 q^{41} -2.61921e13 q^{43} -4.23052e13 q^{45} +1.94586e14 q^{47} +3.29509e14 q^{49} -2.65829e14 q^{51} -4.69769e14 q^{53} -4.90635e14 q^{55} -4.05205e13 q^{57} +1.58789e15 q^{59} -2.16562e15 q^{61} +2.73508e15 q^{63} +1.35083e15 q^{65} +1.54302e15 q^{67} -3.06753e15 q^{69} -4.51083e15 q^{71} -6.24410e15 q^{73} -9.82670e15 q^{75} +3.17201e16 q^{77} -7.95599e14 q^{79} -1.82671e16 q^{81} +2.62699e16 q^{83} +6.23462e15 q^{85} -2.12982e16 q^{87} +3.01291e16 q^{89} -8.73327e16 q^{91} -3.52719e16 q^{93} +9.50348e14 q^{95} +3.38333e16 q^{97} +1.54333e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5880q^{3} + 604044q^{5} + 25350160q^{7} + 449174682q^{9} + O(q^{10}) \) \( 2q - 5880q^{3} + 604044q^{5} + 25350160q^{7} + 449174682q^{9} + 1259648280q^{11} - 1320052580q^{13} - 26621930448q^{15} - 27498226140q^{17} + 101133633832q^{19} + 335430207552q^{21} + 134767491120q^{23} - 448986850114q^{25} - 4619416117680q^{27} + 2337155582652q^{29} + 278836113472q^{31} + 22602380058720q^{33} - 7102242382752q^{35} + 20929802888140q^{37} - 108447997173648q^{39} + 4166592315732q^{41} - 111143148534440q^{43} + 281755357404444q^{45} + 196772651157600q^{47} + 99570326741874q^{49} - 39956666918256q^{51} - 487965122736660q^{53} - 566565363246960q^{55} - 2272313631999840q^{57} + 2835904197813624q^{59} - 67544034994436q^{61} + 3282762121966800q^{63} + 3645157343001768q^{65} - 995171321546360q^{67} - 10188302100105024q^{69} + 3882245493215376q^{71} - 12746425881769580q^{73} - 13688080703624712q^{75} + 31591755846002880q^{77} - 14984271534065504q^{79} + 33380230287529458q^{81} + 43899417809893800q^{83} - 3956216991540264q^{85} - 100892756486768400q^{87} + 51909007958846388q^{89} - 83455169400462112q^{91} - 89807116632334080q^{93} + 101643889304423664q^{95} - 49281007789848380q^{97} + 128223271309133880q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 15636.4 1.37596 0.687982 0.725728i \(-0.258497\pi\)
0.687982 + 0.725728i \(0.258497\pi\)
\(4\) 0 0
\(5\) −366730. −0.419857 −0.209928 0.977717i \(-0.567323\pi\)
−0.209928 + 0.977717i \(0.567323\pi\)
\(6\) 0 0
\(7\) 2.37095e7 1.55449 0.777246 0.629196i \(-0.216616\pi\)
0.777246 + 0.629196i \(0.216616\pi\)
\(8\) 0 0
\(9\) 1.15358e8 0.893277
\(10\) 0 0
\(11\) 1.33786e9 1.88180 0.940902 0.338679i \(-0.109980\pi\)
0.940902 + 0.338679i \(0.109980\pi\)
\(12\) 0 0
\(13\) −3.68345e9 −1.25238 −0.626191 0.779670i \(-0.715387\pi\)
−0.626191 + 0.779670i \(0.715387\pi\)
\(14\) 0 0
\(15\) −5.73434e9 −0.577708
\(16\) 0 0
\(17\) −1.70006e10 −0.591082 −0.295541 0.955330i \(-0.595500\pi\)
−0.295541 + 0.955330i \(0.595500\pi\)
\(18\) 0 0
\(19\) −2.59141e9 −0.0350051 −0.0175025 0.999847i \(-0.505572\pi\)
−0.0175025 + 0.999847i \(0.505572\pi\)
\(20\) 0 0
\(21\) 3.70732e11 2.13893
\(22\) 0 0
\(23\) −1.96178e11 −0.522353 −0.261177 0.965291i \(-0.584111\pi\)
−0.261177 + 0.965291i \(0.584111\pi\)
\(24\) 0 0
\(25\) −6.28449e11 −0.823720
\(26\) 0 0
\(27\) −2.15505e11 −0.146847
\(28\) 0 0
\(29\) −1.36209e12 −0.505618 −0.252809 0.967516i \(-0.581354\pi\)
−0.252809 + 0.967516i \(0.581354\pi\)
\(30\) 0 0
\(31\) −2.25575e12 −0.475025 −0.237512 0.971384i \(-0.576332\pi\)
−0.237512 + 0.971384i \(0.576332\pi\)
\(32\) 0 0
\(33\) 2.09194e13 2.58929
\(34\) 0 0
\(35\) −8.69497e12 −0.652664
\(36\) 0 0
\(37\) 1.28643e12 0.0602105 0.0301052 0.999547i \(-0.490416\pi\)
0.0301052 + 0.999547i \(0.490416\pi\)
\(38\) 0 0
\(39\) −5.75961e13 −1.72323
\(40\) 0 0
\(41\) 3.74268e13 0.732014 0.366007 0.930612i \(-0.380725\pi\)
0.366007 + 0.930612i \(0.380725\pi\)
\(42\) 0 0
\(43\) −2.61921e13 −0.341734 −0.170867 0.985294i \(-0.554657\pi\)
−0.170867 + 0.985294i \(0.554657\pi\)
\(44\) 0 0
\(45\) −4.23052e13 −0.375048
\(46\) 0 0
\(47\) 1.94586e14 1.19201 0.596006 0.802980i \(-0.296754\pi\)
0.596006 + 0.802980i \(0.296754\pi\)
\(48\) 0 0
\(49\) 3.29509e14 1.41645
\(50\) 0 0
\(51\) −2.65829e14 −0.813308
\(52\) 0 0
\(53\) −4.69769e14 −1.03643 −0.518214 0.855251i \(-0.673403\pi\)
−0.518214 + 0.855251i \(0.673403\pi\)
\(54\) 0 0
\(55\) −4.90635e14 −0.790088
\(56\) 0 0
\(57\) −4.05205e13 −0.0481657
\(58\) 0 0
\(59\) 1.58789e15 1.40792 0.703962 0.710238i \(-0.251413\pi\)
0.703962 + 0.710238i \(0.251413\pi\)
\(60\) 0 0
\(61\) −2.16562e15 −1.44637 −0.723183 0.690656i \(-0.757322\pi\)
−0.723183 + 0.690656i \(0.757322\pi\)
\(62\) 0 0
\(63\) 2.73508e15 1.38859
\(64\) 0 0
\(65\) 1.35083e15 0.525821
\(66\) 0 0
\(67\) 1.54302e15 0.464232 0.232116 0.972688i \(-0.425435\pi\)
0.232116 + 0.972688i \(0.425435\pi\)
\(68\) 0 0
\(69\) −3.06753e15 −0.718739
\(70\) 0 0
\(71\) −4.51083e15 −0.829011 −0.414506 0.910047i \(-0.636046\pi\)
−0.414506 + 0.910047i \(0.636046\pi\)
\(72\) 0 0
\(73\) −6.24410e15 −0.906203 −0.453101 0.891459i \(-0.649682\pi\)
−0.453101 + 0.891459i \(0.649682\pi\)
\(74\) 0 0
\(75\) −9.82670e15 −1.13341
\(76\) 0 0
\(77\) 3.17201e16 2.92525
\(78\) 0 0
\(79\) −7.95599e14 −0.0590016 −0.0295008 0.999565i \(-0.509392\pi\)
−0.0295008 + 0.999565i \(0.509392\pi\)
\(80\) 0 0
\(81\) −1.82671e16 −1.09533
\(82\) 0 0
\(83\) 2.62699e16 1.28025 0.640125 0.768271i \(-0.278883\pi\)
0.640125 + 0.768271i \(0.278883\pi\)
\(84\) 0 0
\(85\) 6.23462e15 0.248170
\(86\) 0 0
\(87\) −2.12982e16 −0.695712
\(88\) 0 0
\(89\) 3.01291e16 0.811280 0.405640 0.914033i \(-0.367049\pi\)
0.405640 + 0.914033i \(0.367049\pi\)
\(90\) 0 0
\(91\) −8.73327e16 −1.94682
\(92\) 0 0
\(93\) −3.52719e16 −0.653617
\(94\) 0 0
\(95\) 9.50348e14 0.0146971
\(96\) 0 0
\(97\) 3.38333e16 0.438314 0.219157 0.975690i \(-0.429669\pi\)
0.219157 + 0.975690i \(0.429669\pi\)
\(98\) 0 0
\(99\) 1.54333e17 1.68097
\(100\) 0 0
\(101\) 5.50280e16 0.505653 0.252826 0.967512i \(-0.418640\pi\)
0.252826 + 0.967512i \(0.418640\pi\)
\(102\) 0 0
\(103\) −2.99690e16 −0.233107 −0.116554 0.993184i \(-0.537185\pi\)
−0.116554 + 0.993184i \(0.537185\pi\)
\(104\) 0 0
\(105\) −1.35958e17 −0.898042
\(106\) 0 0
\(107\) 1.83477e17 1.03233 0.516166 0.856489i \(-0.327359\pi\)
0.516166 + 0.856489i \(0.327359\pi\)
\(108\) 0 0
\(109\) −2.65387e17 −1.27572 −0.637859 0.770153i \(-0.720180\pi\)
−0.637859 + 0.770153i \(0.720180\pi\)
\(110\) 0 0
\(111\) 2.01152e16 0.0828475
\(112\) 0 0
\(113\) −1.05318e17 −0.372681 −0.186341 0.982485i \(-0.559663\pi\)
−0.186341 + 0.982485i \(0.559663\pi\)
\(114\) 0 0
\(115\) 7.19444e16 0.219313
\(116\) 0 0
\(117\) −4.24915e17 −1.11872
\(118\) 0 0
\(119\) −4.03075e17 −0.918833
\(120\) 0 0
\(121\) 1.28444e18 2.54119
\(122\) 0 0
\(123\) 5.85221e17 1.00723
\(124\) 0 0
\(125\) 5.10263e17 0.765701
\(126\) 0 0
\(127\) −7.92218e17 −1.03875 −0.519377 0.854545i \(-0.673836\pi\)
−0.519377 + 0.854545i \(0.673836\pi\)
\(128\) 0 0
\(129\) −4.09551e17 −0.470213
\(130\) 0 0
\(131\) −7.17088e17 −0.722381 −0.361191 0.932492i \(-0.617630\pi\)
−0.361191 + 0.932492i \(0.617630\pi\)
\(132\) 0 0
\(133\) −6.14411e16 −0.0544152
\(134\) 0 0
\(135\) 7.90322e16 0.0616548
\(136\) 0 0
\(137\) −1.94843e18 −1.34141 −0.670703 0.741726i \(-0.734008\pi\)
−0.670703 + 0.741726i \(0.734008\pi\)
\(138\) 0 0
\(139\) 1.23071e18 0.749085 0.374543 0.927210i \(-0.377800\pi\)
0.374543 + 0.927210i \(0.377800\pi\)
\(140\) 0 0
\(141\) 3.04264e18 1.64016
\(142\) 0 0
\(143\) −4.92796e18 −2.35674
\(144\) 0 0
\(145\) 4.99518e17 0.212287
\(146\) 0 0
\(147\) 5.15235e18 1.94898
\(148\) 0 0
\(149\) 2.33643e18 0.787898 0.393949 0.919132i \(-0.371109\pi\)
0.393949 + 0.919132i \(0.371109\pi\)
\(150\) 0 0
\(151\) 5.57631e17 0.167897 0.0839485 0.996470i \(-0.473247\pi\)
0.0839485 + 0.996470i \(0.473247\pi\)
\(152\) 0 0
\(153\) −1.96115e18 −0.528000
\(154\) 0 0
\(155\) 8.27250e17 0.199442
\(156\) 0 0
\(157\) −5.14946e18 −1.11331 −0.556653 0.830745i \(-0.687915\pi\)
−0.556653 + 0.830745i \(0.687915\pi\)
\(158\) 0 0
\(159\) −7.34551e18 −1.42609
\(160\) 0 0
\(161\) −4.65129e18 −0.811994
\(162\) 0 0
\(163\) 1.04483e19 1.64230 0.821148 0.570715i \(-0.193334\pi\)
0.821148 + 0.570715i \(0.193334\pi\)
\(164\) 0 0
\(165\) −7.67178e18 −1.08713
\(166\) 0 0
\(167\) 1.10713e19 1.41614 0.708071 0.706141i \(-0.249565\pi\)
0.708071 + 0.706141i \(0.249565\pi\)
\(168\) 0 0
\(169\) 4.91741e18 0.568459
\(170\) 0 0
\(171\) −2.98940e17 −0.0312692
\(172\) 0 0
\(173\) −1.38307e18 −0.131055 −0.0655274 0.997851i \(-0.520873\pi\)
−0.0655274 + 0.997851i \(0.520873\pi\)
\(174\) 0 0
\(175\) −1.49002e19 −1.28047
\(176\) 0 0
\(177\) 2.48290e19 1.93725
\(178\) 0 0
\(179\) 1.50209e19 1.06524 0.532618 0.846356i \(-0.321208\pi\)
0.532618 + 0.846356i \(0.321208\pi\)
\(180\) 0 0
\(181\) −3.60783e17 −0.0232798 −0.0116399 0.999932i \(-0.503705\pi\)
−0.0116399 + 0.999932i \(0.503705\pi\)
\(182\) 0 0
\(183\) −3.38625e19 −1.99015
\(184\) 0 0
\(185\) −4.71773e17 −0.0252798
\(186\) 0 0
\(187\) −2.27445e19 −1.11230
\(188\) 0 0
\(189\) −5.10952e18 −0.228273
\(190\) 0 0
\(191\) 8.97904e18 0.366814 0.183407 0.983037i \(-0.441287\pi\)
0.183407 + 0.983037i \(0.441287\pi\)
\(192\) 0 0
\(193\) −2.87785e19 −1.07605 −0.538023 0.842930i \(-0.680829\pi\)
−0.538023 + 0.842930i \(0.680829\pi\)
\(194\) 0 0
\(195\) 2.11222e19 0.723510
\(196\) 0 0
\(197\) 5.29790e19 1.66395 0.831977 0.554811i \(-0.187209\pi\)
0.831977 + 0.554811i \(0.187209\pi\)
\(198\) 0 0
\(199\) 2.52925e19 0.729021 0.364510 0.931199i \(-0.381236\pi\)
0.364510 + 0.931199i \(0.381236\pi\)
\(200\) 0 0
\(201\) 2.41273e19 0.638767
\(202\) 0 0
\(203\) −3.22944e19 −0.785979
\(204\) 0 0
\(205\) −1.37255e19 −0.307341
\(206\) 0 0
\(207\) −2.26307e19 −0.466606
\(208\) 0 0
\(209\) −3.46696e18 −0.0658727
\(210\) 0 0
\(211\) 6.87338e19 1.20440 0.602198 0.798346i \(-0.294292\pi\)
0.602198 + 0.798346i \(0.294292\pi\)
\(212\) 0 0
\(213\) −7.05333e19 −1.14069
\(214\) 0 0
\(215\) 9.60541e18 0.143479
\(216\) 0 0
\(217\) −5.34826e19 −0.738423
\(218\) 0 0
\(219\) −9.76354e19 −1.24690
\(220\) 0 0
\(221\) 6.26208e19 0.740260
\(222\) 0 0
\(223\) −4.40564e18 −0.0482412 −0.0241206 0.999709i \(-0.507679\pi\)
−0.0241206 + 0.999709i \(0.507679\pi\)
\(224\) 0 0
\(225\) −7.24965e19 −0.735810
\(226\) 0 0
\(227\) 3.27911e19 0.308700 0.154350 0.988016i \(-0.450672\pi\)
0.154350 + 0.988016i \(0.450672\pi\)
\(228\) 0 0
\(229\) 9.54497e19 0.834013 0.417006 0.908904i \(-0.363079\pi\)
0.417006 + 0.908904i \(0.363079\pi\)
\(230\) 0 0
\(231\) 4.95989e20 4.02504
\(232\) 0 0
\(233\) −2.07634e20 −1.56593 −0.782965 0.622066i \(-0.786294\pi\)
−0.782965 + 0.622066i \(0.786294\pi\)
\(234\) 0 0
\(235\) −7.13605e19 −0.500474
\(236\) 0 0
\(237\) −1.24403e19 −0.0811841
\(238\) 0 0
\(239\) −2.76425e20 −1.67956 −0.839780 0.542927i \(-0.817316\pi\)
−0.839780 + 0.542927i \(0.817316\pi\)
\(240\) 0 0
\(241\) 2.35403e20 1.33250 0.666249 0.745730i \(-0.267899\pi\)
0.666249 + 0.745730i \(0.267899\pi\)
\(242\) 0 0
\(243\) −2.57802e20 −1.36029
\(244\) 0 0
\(245\) −1.20841e20 −0.594705
\(246\) 0 0
\(247\) 9.54535e18 0.0438397
\(248\) 0 0
\(249\) 4.10768e20 1.76158
\(250\) 0 0
\(251\) −3.91593e20 −1.56895 −0.784473 0.620163i \(-0.787067\pi\)
−0.784473 + 0.620163i \(0.787067\pi\)
\(252\) 0 0
\(253\) −2.62460e20 −0.982967
\(254\) 0 0
\(255\) 9.74872e19 0.341473
\(256\) 0 0
\(257\) 4.14828e20 1.35968 0.679839 0.733361i \(-0.262049\pi\)
0.679839 + 0.733361i \(0.262049\pi\)
\(258\) 0 0
\(259\) 3.05006e19 0.0935968
\(260\) 0 0
\(261\) −1.57128e20 −0.451657
\(262\) 0 0
\(263\) 9.82407e19 0.264647 0.132324 0.991207i \(-0.457756\pi\)
0.132324 + 0.991207i \(0.457756\pi\)
\(264\) 0 0
\(265\) 1.72278e20 0.435151
\(266\) 0 0
\(267\) 4.71112e20 1.11629
\(268\) 0 0
\(269\) 5.22922e20 1.16290 0.581450 0.813582i \(-0.302486\pi\)
0.581450 + 0.813582i \(0.302486\pi\)
\(270\) 0 0
\(271\) 4.47142e20 0.933698 0.466849 0.884337i \(-0.345389\pi\)
0.466849 + 0.884337i \(0.345389\pi\)
\(272\) 0 0
\(273\) −1.36557e21 −2.67875
\(274\) 0 0
\(275\) −8.40780e20 −1.55008
\(276\) 0 0
\(277\) −1.36272e20 −0.236226 −0.118113 0.993000i \(-0.537684\pi\)
−0.118113 + 0.993000i \(0.537684\pi\)
\(278\) 0 0
\(279\) −2.60219e20 −0.424329
\(280\) 0 0
\(281\) 1.26390e20 0.193958 0.0969790 0.995286i \(-0.469082\pi\)
0.0969790 + 0.995286i \(0.469082\pi\)
\(282\) 0 0
\(283\) −9.81664e19 −0.141833 −0.0709167 0.997482i \(-0.522592\pi\)
−0.0709167 + 0.997482i \(0.522592\pi\)
\(284\) 0 0
\(285\) 1.48601e19 0.0202227
\(286\) 0 0
\(287\) 8.87370e20 1.13791
\(288\) 0 0
\(289\) −5.38220e20 −0.650622
\(290\) 0 0
\(291\) 5.29033e20 0.603104
\(292\) 0 0
\(293\) 1.01600e21 1.09275 0.546374 0.837541i \(-0.316008\pi\)
0.546374 + 0.837541i \(0.316008\pi\)
\(294\) 0 0
\(295\) −5.82327e20 −0.591126
\(296\) 0 0
\(297\) −2.88317e20 −0.276338
\(298\) 0 0
\(299\) 7.22613e20 0.654185
\(300\) 0 0
\(301\) −6.21000e20 −0.531223
\(302\) 0 0
\(303\) 8.60442e20 0.695760
\(304\) 0 0
\(305\) 7.94196e20 0.607266
\(306\) 0 0
\(307\) 1.04896e21 0.758720 0.379360 0.925249i \(-0.376144\pi\)
0.379360 + 0.925249i \(0.376144\pi\)
\(308\) 0 0
\(309\) −4.68609e20 −0.320747
\(310\) 0 0
\(311\) −1.73873e21 −1.12660 −0.563298 0.826254i \(-0.690468\pi\)
−0.563298 + 0.826254i \(0.690468\pi\)
\(312\) 0 0
\(313\) −2.56401e21 −1.57323 −0.786616 0.617443i \(-0.788169\pi\)
−0.786616 + 0.617443i \(0.788169\pi\)
\(314\) 0 0
\(315\) −1.00303e21 −0.583010
\(316\) 0 0
\(317\) −3.06492e20 −0.168817 −0.0844084 0.996431i \(-0.526900\pi\)
−0.0844084 + 0.996431i \(0.526900\pi\)
\(318\) 0 0
\(319\) −1.82229e21 −0.951474
\(320\) 0 0
\(321\) 2.86892e21 1.42045
\(322\) 0 0
\(323\) 4.40556e19 0.0206909
\(324\) 0 0
\(325\) 2.31486e21 1.03161
\(326\) 0 0
\(327\) −4.14971e21 −1.75534
\(328\) 0 0
\(329\) 4.61354e21 1.85297
\(330\) 0 0
\(331\) 4.66802e21 1.78072 0.890359 0.455259i \(-0.150453\pi\)
0.890359 + 0.455259i \(0.150453\pi\)
\(332\) 0 0
\(333\) 1.48400e20 0.0537846
\(334\) 0 0
\(335\) −5.65871e20 −0.194911
\(336\) 0 0
\(337\) −1.64491e21 −0.538626 −0.269313 0.963053i \(-0.586797\pi\)
−0.269313 + 0.963053i \(0.586797\pi\)
\(338\) 0 0
\(339\) −1.64681e21 −0.512796
\(340\) 0 0
\(341\) −3.01789e21 −0.893904
\(342\) 0 0
\(343\) 2.29694e21 0.647365
\(344\) 0 0
\(345\) 1.12495e21 0.301767
\(346\) 0 0
\(347\) −9.19804e20 −0.234906 −0.117453 0.993078i \(-0.537473\pi\)
−0.117453 + 0.993078i \(0.537473\pi\)
\(348\) 0 0
\(349\) −1.23910e21 −0.301363 −0.150681 0.988582i \(-0.548147\pi\)
−0.150681 + 0.988582i \(0.548147\pi\)
\(350\) 0 0
\(351\) 7.93804e20 0.183909
\(352\) 0 0
\(353\) −2.26089e21 −0.499109 −0.249554 0.968361i \(-0.580284\pi\)
−0.249554 + 0.968361i \(0.580284\pi\)
\(354\) 0 0
\(355\) 1.65426e21 0.348066
\(356\) 0 0
\(357\) −6.30266e21 −1.26428
\(358\) 0 0
\(359\) −9.96795e21 −1.90679 −0.953395 0.301725i \(-0.902438\pi\)
−0.953395 + 0.301725i \(0.902438\pi\)
\(360\) 0 0
\(361\) −5.47367e21 −0.998775
\(362\) 0 0
\(363\) 2.00840e22 3.49658
\(364\) 0 0
\(365\) 2.28990e21 0.380475
\(366\) 0 0
\(367\) 5.17139e21 0.820248 0.410124 0.912030i \(-0.365485\pi\)
0.410124 + 0.912030i \(0.365485\pi\)
\(368\) 0 0
\(369\) 4.31748e21 0.653892
\(370\) 0 0
\(371\) −1.11380e22 −1.61112
\(372\) 0 0
\(373\) 1.06342e22 1.46954 0.734769 0.678317i \(-0.237290\pi\)
0.734769 + 0.678317i \(0.237290\pi\)
\(374\) 0 0
\(375\) 7.97870e21 1.05358
\(376\) 0 0
\(377\) 5.01719e21 0.633226
\(378\) 0 0
\(379\) −6.81105e21 −0.821827 −0.410914 0.911674i \(-0.634790\pi\)
−0.410914 + 0.911674i \(0.634790\pi\)
\(380\) 0 0
\(381\) −1.23875e22 −1.42929
\(382\) 0 0
\(383\) −6.65462e21 −0.734402 −0.367201 0.930142i \(-0.619684\pi\)
−0.367201 + 0.930142i \(0.619684\pi\)
\(384\) 0 0
\(385\) −1.16327e22 −1.22819
\(386\) 0 0
\(387\) −3.02146e21 −0.305263
\(388\) 0 0
\(389\) 1.61442e22 1.56115 0.780574 0.625063i \(-0.214927\pi\)
0.780574 + 0.625063i \(0.214927\pi\)
\(390\) 0 0
\(391\) 3.33515e21 0.308754
\(392\) 0 0
\(393\) −1.12127e22 −0.993971
\(394\) 0 0
\(395\) 2.91770e20 0.0247722
\(396\) 0 0
\(397\) 1.02258e21 0.0831725 0.0415862 0.999135i \(-0.486759\pi\)
0.0415862 + 0.999135i \(0.486759\pi\)
\(398\) 0 0
\(399\) −9.60719e20 −0.0748733
\(400\) 0 0
\(401\) −1.32825e22 −0.992095 −0.496047 0.868295i \(-0.665216\pi\)
−0.496047 + 0.868295i \(0.665216\pi\)
\(402\) 0 0
\(403\) 8.30895e21 0.594912
\(404\) 0 0
\(405\) 6.69908e21 0.459883
\(406\) 0 0
\(407\) 1.72107e21 0.113304
\(408\) 0 0
\(409\) 2.58540e22 1.63260 0.816300 0.577628i \(-0.196022\pi\)
0.816300 + 0.577628i \(0.196022\pi\)
\(410\) 0 0
\(411\) −3.04665e22 −1.84573
\(412\) 0 0
\(413\) 3.76481e22 2.18861
\(414\) 0 0
\(415\) −9.63395e21 −0.537521
\(416\) 0 0
\(417\) 1.92440e22 1.03071
\(418\) 0 0
\(419\) 1.13941e22 0.585951 0.292975 0.956120i \(-0.405355\pi\)
0.292975 + 0.956120i \(0.405355\pi\)
\(420\) 0 0
\(421\) −2.96045e22 −1.46204 −0.731021 0.682354i \(-0.760956\pi\)
−0.731021 + 0.682354i \(0.760956\pi\)
\(422\) 0 0
\(423\) 2.24471e22 1.06480
\(424\) 0 0
\(425\) 1.06840e22 0.486887
\(426\) 0 0
\(427\) −5.13457e22 −2.24837
\(428\) 0 0
\(429\) −7.70557e22 −3.24278
\(430\) 0 0
\(431\) 1.85742e22 0.751369 0.375684 0.926748i \(-0.377408\pi\)
0.375684 + 0.926748i \(0.377408\pi\)
\(432\) 0 0
\(433\) 1.71171e21 0.0665707 0.0332854 0.999446i \(-0.489403\pi\)
0.0332854 + 0.999446i \(0.489403\pi\)
\(434\) 0 0
\(435\) 7.81068e21 0.292099
\(436\) 0 0
\(437\) 5.08379e20 0.0182850
\(438\) 0 0
\(439\) 3.04484e22 1.05346 0.526728 0.850034i \(-0.323419\pi\)
0.526728 + 0.850034i \(0.323419\pi\)
\(440\) 0 0
\(441\) 3.80115e22 1.26528
\(442\) 0 0
\(443\) −2.23180e22 −0.714864 −0.357432 0.933939i \(-0.616348\pi\)
−0.357432 + 0.933939i \(0.616348\pi\)
\(444\) 0 0
\(445\) −1.10492e22 −0.340621
\(446\) 0 0
\(447\) 3.65335e22 1.08412
\(448\) 0 0
\(449\) −5.77731e22 −1.65056 −0.825281 0.564723i \(-0.808983\pi\)
−0.825281 + 0.564723i \(0.808983\pi\)
\(450\) 0 0
\(451\) 5.00720e22 1.37751
\(452\) 0 0
\(453\) 8.71936e21 0.231020
\(454\) 0 0
\(455\) 3.20275e22 0.817384
\(456\) 0 0
\(457\) −4.47877e22 −1.10121 −0.550606 0.834766i \(-0.685603\pi\)
−0.550606 + 0.834766i \(0.685603\pi\)
\(458\) 0 0
\(459\) 3.66372e21 0.0867989
\(460\) 0 0
\(461\) 7.44258e22 1.69928 0.849641 0.527361i \(-0.176818\pi\)
0.849641 + 0.527361i \(0.176818\pi\)
\(462\) 0 0
\(463\) −3.47065e22 −0.763787 −0.381893 0.924206i \(-0.624728\pi\)
−0.381893 + 0.924206i \(0.624728\pi\)
\(464\) 0 0
\(465\) 1.29352e22 0.274425
\(466\) 0 0
\(467\) −3.07656e22 −0.629321 −0.314661 0.949204i \(-0.601891\pi\)
−0.314661 + 0.949204i \(0.601891\pi\)
\(468\) 0 0
\(469\) 3.65842e22 0.721646
\(470\) 0 0
\(471\) −8.05192e22 −1.53187
\(472\) 0 0
\(473\) −3.50414e22 −0.643076
\(474\) 0 0
\(475\) 1.62857e21 0.0288344
\(476\) 0 0
\(477\) −5.41916e22 −0.925817
\(478\) 0 0
\(479\) −2.50857e22 −0.413595 −0.206797 0.978384i \(-0.566304\pi\)
−0.206797 + 0.978384i \(0.566304\pi\)
\(480\) 0 0
\(481\) −4.73851e21 −0.0754065
\(482\) 0 0
\(483\) −7.27295e22 −1.11728
\(484\) 0 0
\(485\) −1.24077e22 −0.184029
\(486\) 0 0
\(487\) 1.25508e23 1.79752 0.898761 0.438439i \(-0.144468\pi\)
0.898761 + 0.438439i \(0.144468\pi\)
\(488\) 0 0
\(489\) 1.63374e23 2.25974
\(490\) 0 0
\(491\) 5.29654e22 0.707619 0.353809 0.935318i \(-0.384886\pi\)
0.353809 + 0.935318i \(0.384886\pi\)
\(492\) 0 0
\(493\) 2.31563e22 0.298862
\(494\) 0 0
\(495\) −5.65986e22 −0.705767
\(496\) 0 0
\(497\) −1.06949e23 −1.28869
\(498\) 0 0
\(499\) 2.81497e22 0.327807 0.163904 0.986476i \(-0.447591\pi\)
0.163904 + 0.986476i \(0.447591\pi\)
\(500\) 0 0
\(501\) 1.73115e23 1.94856
\(502\) 0 0
\(503\) −5.42160e22 −0.589929 −0.294964 0.955508i \(-0.595308\pi\)
−0.294964 + 0.955508i \(0.595308\pi\)
\(504\) 0 0
\(505\) −2.01804e22 −0.212302
\(506\) 0 0
\(507\) 7.68907e22 0.782179
\(508\) 0 0
\(509\) −1.78224e23 −1.75333 −0.876666 0.481099i \(-0.840238\pi\)
−0.876666 + 0.481099i \(0.840238\pi\)
\(510\) 0 0
\(511\) −1.48044e23 −1.40869
\(512\) 0 0
\(513\) 5.58463e20 0.00514040
\(514\) 0 0
\(515\) 1.09905e22 0.0978717
\(516\) 0 0
\(517\) 2.60330e23 2.24313
\(518\) 0 0
\(519\) −2.16263e22 −0.180327
\(520\) 0 0
\(521\) −1.94263e22 −0.156773 −0.0783863 0.996923i \(-0.524977\pi\)
−0.0783863 + 0.996923i \(0.524977\pi\)
\(522\) 0 0
\(523\) −1.65768e23 −1.29490 −0.647451 0.762107i \(-0.724165\pi\)
−0.647451 + 0.762107i \(0.724165\pi\)
\(524\) 0 0
\(525\) −2.32986e23 −1.76188
\(526\) 0 0
\(527\) 3.83491e22 0.280779
\(528\) 0 0
\(529\) −1.02564e23 −0.727147
\(530\) 0 0
\(531\) 1.83176e23 1.25767
\(532\) 0 0
\(533\) −1.37860e23 −0.916761
\(534\) 0 0
\(535\) −6.72864e22 −0.433431
\(536\) 0 0
\(537\) 2.34874e23 1.46573
\(538\) 0 0
\(539\) 4.40839e23 2.66548
\(540\) 0 0
\(541\) 2.94676e22 0.172650 0.0863251 0.996267i \(-0.472488\pi\)
0.0863251 + 0.996267i \(0.472488\pi\)
\(542\) 0 0
\(543\) −5.64137e21 −0.0320321
\(544\) 0 0
\(545\) 9.73253e22 0.535618
\(546\) 0 0
\(547\) 8.00474e22 0.427027 0.213513 0.976940i \(-0.431509\pi\)
0.213513 + 0.976940i \(0.431509\pi\)
\(548\) 0 0
\(549\) −2.49821e23 −1.29201
\(550\) 0 0
\(551\) 3.52974e21 0.0176992
\(552\) 0 0
\(553\) −1.88632e22 −0.0917176
\(554\) 0 0
\(555\) −7.37684e21 −0.0347840
\(556\) 0 0
\(557\) −1.60669e23 −0.734790 −0.367395 0.930065i \(-0.619750\pi\)
−0.367395 + 0.930065i \(0.619750\pi\)
\(558\) 0 0
\(559\) 9.64772e22 0.427981
\(560\) 0 0
\(561\) −3.55643e23 −1.53049
\(562\) 0 0
\(563\) −9.05044e22 −0.377875 −0.188937 0.981989i \(-0.560504\pi\)
−0.188937 + 0.981989i \(0.560504\pi\)
\(564\) 0 0
\(565\) 3.86234e22 0.156473
\(566\) 0 0
\(567\) −4.33103e23 −1.70269
\(568\) 0 0
\(569\) 4.23350e23 1.61527 0.807635 0.589683i \(-0.200747\pi\)
0.807635 + 0.589683i \(0.200747\pi\)
\(570\) 0 0
\(571\) 1.34739e21 0.00498985 0.00249493 0.999997i \(-0.499206\pi\)
0.00249493 + 0.999997i \(0.499206\pi\)
\(572\) 0 0
\(573\) 1.40400e23 0.504723
\(574\) 0 0
\(575\) 1.23288e23 0.430273
\(576\) 0 0
\(577\) −4.74144e22 −0.160663 −0.0803315 0.996768i \(-0.525598\pi\)
−0.0803315 + 0.996768i \(0.525598\pi\)
\(578\) 0 0
\(579\) −4.49993e23 −1.48060
\(580\) 0 0
\(581\) 6.22846e23 1.99014
\(582\) 0 0
\(583\) −6.28487e23 −1.95036
\(584\) 0 0
\(585\) 1.55829e23 0.469703
\(586\) 0 0
\(587\) 1.06143e23 0.310790 0.155395 0.987852i \(-0.450335\pi\)
0.155395 + 0.987852i \(0.450335\pi\)
\(588\) 0 0
\(589\) 5.84558e21 0.0166283
\(590\) 0 0
\(591\) 8.28403e23 2.28954
\(592\) 0 0
\(593\) −3.00230e23 −0.806285 −0.403142 0.915137i \(-0.632082\pi\)
−0.403142 + 0.915137i \(0.632082\pi\)
\(594\) 0 0
\(595\) 1.47820e23 0.385778
\(596\) 0 0
\(597\) 3.95484e23 1.00311
\(598\) 0 0
\(599\) 1.55455e23 0.383246 0.191623 0.981469i \(-0.438625\pi\)
0.191623 + 0.981469i \(0.438625\pi\)
\(600\) 0 0
\(601\) −7.11005e22 −0.170388 −0.0851941 0.996364i \(-0.527151\pi\)
−0.0851941 + 0.996364i \(0.527151\pi\)
\(602\) 0 0
\(603\) 1.77999e23 0.414688
\(604\) 0 0
\(605\) −4.71041e23 −1.06693
\(606\) 0 0
\(607\) −2.07763e23 −0.457576 −0.228788 0.973476i \(-0.573476\pi\)
−0.228788 + 0.973476i \(0.573476\pi\)
\(608\) 0 0
\(609\) −5.04970e23 −1.08148
\(610\) 0 0
\(611\) −7.16749e23 −1.49285
\(612\) 0 0
\(613\) 5.47051e23 1.10819 0.554095 0.832454i \(-0.313064\pi\)
0.554095 + 0.832454i \(0.313064\pi\)
\(614\) 0 0
\(615\) −2.14618e23 −0.422890
\(616\) 0 0
\(617\) −5.79656e23 −1.11108 −0.555542 0.831489i \(-0.687489\pi\)
−0.555542 + 0.831489i \(0.687489\pi\)
\(618\) 0 0
\(619\) −5.83066e23 −1.08729 −0.543647 0.839314i \(-0.682957\pi\)
−0.543647 + 0.839314i \(0.682957\pi\)
\(620\) 0 0
\(621\) 4.22775e22 0.0767062
\(622\) 0 0
\(623\) 7.14345e23 1.26113
\(624\) 0 0
\(625\) 2.92340e23 0.502236
\(626\) 0 0
\(627\) −5.42109e22 −0.0906385
\(628\) 0 0
\(629\) −2.18701e22 −0.0355894
\(630\) 0 0
\(631\) 9.51360e23 1.50694 0.753469 0.657483i \(-0.228379\pi\)
0.753469 + 0.657483i \(0.228379\pi\)
\(632\) 0 0
\(633\) 1.07475e24 1.65721
\(634\) 0 0
\(635\) 2.90530e23 0.436128
\(636\) 0 0
\(637\) −1.21373e24 −1.77393
\(638\) 0 0
\(639\) −5.20360e23 −0.740537
\(640\) 0 0
\(641\) −9.77138e23 −1.35414 −0.677069 0.735920i \(-0.736750\pi\)
−0.677069 + 0.735920i \(0.736750\pi\)
\(642\) 0 0
\(643\) 5.56439e23 0.750973 0.375486 0.926828i \(-0.377476\pi\)
0.375486 + 0.926828i \(0.377476\pi\)
\(644\) 0 0
\(645\) 1.50194e23 0.197422
\(646\) 0 0
\(647\) 1.15984e24 1.48495 0.742477 0.669871i \(-0.233651\pi\)
0.742477 + 0.669871i \(0.233651\pi\)
\(648\) 0 0
\(649\) 2.12439e24 2.64944
\(650\) 0 0
\(651\) −8.36278e23 −1.01604
\(652\) 0 0
\(653\) −9.87432e23 −1.16881 −0.584407 0.811461i \(-0.698673\pi\)
−0.584407 + 0.811461i \(0.698673\pi\)
\(654\) 0 0
\(655\) 2.62977e23 0.303297
\(656\) 0 0
\(657\) −7.20306e23 −0.809490
\(658\) 0 0
\(659\) −4.97449e23 −0.544782 −0.272391 0.962187i \(-0.587814\pi\)
−0.272391 + 0.962187i \(0.587814\pi\)
\(660\) 0 0
\(661\) −1.16033e24 −1.23843 −0.619214 0.785222i \(-0.712549\pi\)
−0.619214 + 0.785222i \(0.712549\pi\)
\(662\) 0 0
\(663\) 9.79167e23 1.01857
\(664\) 0 0
\(665\) 2.25323e22 0.0228466
\(666\) 0 0
\(667\) 2.67212e23 0.264111
\(668\) 0 0
\(669\) −6.88886e22 −0.0663781
\(670\) 0 0
\(671\) −2.89730e24 −2.72178
\(672\) 0 0
\(673\) −1.13175e24 −1.03663 −0.518315 0.855190i \(-0.673441\pi\)
−0.518315 + 0.855190i \(0.673441\pi\)
\(674\) 0 0
\(675\) 1.35434e23 0.120961
\(676\) 0 0
\(677\) 1.09418e24 0.952983 0.476492 0.879179i \(-0.341908\pi\)
0.476492 + 0.879179i \(0.341908\pi\)
\(678\) 0 0
\(679\) 8.02171e23 0.681356
\(680\) 0 0
\(681\) 5.12736e23 0.424760
\(682\) 0 0
\(683\) 1.05400e22 0.00851660 0.00425830 0.999991i \(-0.498645\pi\)
0.00425830 + 0.999991i \(0.498645\pi\)
\(684\) 0 0
\(685\) 7.14548e23 0.563198
\(686\) 0 0
\(687\) 1.49249e24 1.14757
\(688\) 0 0
\(689\) 1.73037e24 1.29800
\(690\) 0 0
\(691\) −1.61636e24 −1.18297 −0.591485 0.806316i \(-0.701458\pi\)
−0.591485 + 0.806316i \(0.701458\pi\)
\(692\) 0 0
\(693\) 3.65916e24 2.61306
\(694\) 0 0
\(695\) −4.51339e23 −0.314508
\(696\) 0 0
\(697\) −6.36277e23 −0.432681
\(698\) 0 0
\(699\) −3.24665e24 −2.15466
\(700\) 0 0
\(701\) 1.86841e23 0.121024 0.0605118 0.998167i \(-0.480727\pi\)
0.0605118 + 0.998167i \(0.480727\pi\)
\(702\) 0 0
\(703\) −3.33368e21 −0.00210767
\(704\) 0 0
\(705\) −1.11582e24 −0.688634
\(706\) 0 0
\(707\) 1.30469e24 0.786033
\(708\) 0 0
\(709\) −2.18773e23 −0.128677 −0.0643385 0.997928i \(-0.520494\pi\)
−0.0643385 + 0.997928i \(0.520494\pi\)
\(710\) 0 0
\(711\) −9.17786e22 −0.0527048
\(712\) 0 0
\(713\) 4.42529e23 0.248131
\(714\) 0 0
\(715\) 1.80723e24 0.989491
\(716\) 0 0
\(717\) −4.32230e24 −2.31101
\(718\) 0 0
\(719\) −1.15548e24 −0.603345 −0.301673 0.953412i \(-0.597545\pi\)
−0.301673 + 0.953412i \(0.597545\pi\)
\(720\) 0 0
\(721\) −7.10550e23 −0.362364
\(722\) 0 0
\(723\) 3.68086e24 1.83347
\(724\) 0 0
\(725\) 8.56003e23 0.416488
\(726\) 0 0
\(727\) −3.57207e24 −1.69777 −0.848883 0.528581i \(-0.822724\pi\)
−0.848883 + 0.528581i \(0.822724\pi\)
\(728\) 0 0
\(729\) −1.67208e24 −0.776379
\(730\) 0 0
\(731\) 4.45280e23 0.201993
\(732\) 0 0
\(733\) −4.90391e23 −0.217350 −0.108675 0.994077i \(-0.534661\pi\)
−0.108675 + 0.994077i \(0.534661\pi\)
\(734\) 0 0
\(735\) −1.88952e24 −0.818293
\(736\) 0 0
\(737\) 2.06435e24 0.873594
\(738\) 0 0
\(739\) 2.17372e24 0.898928 0.449464 0.893298i \(-0.351615\pi\)
0.449464 + 0.893298i \(0.351615\pi\)
\(740\) 0 0
\(741\) 1.49255e23 0.0603219
\(742\) 0 0
\(743\) 3.08963e24 1.22040 0.610199 0.792248i \(-0.291089\pi\)
0.610199 + 0.792248i \(0.291089\pi\)
\(744\) 0 0
\(745\) −8.56839e23 −0.330804
\(746\) 0 0
\(747\) 3.03044e24 1.14362
\(748\) 0 0
\(749\) 4.35014e24 1.60475
\(750\) 0 0
\(751\) 4.58172e24 1.65230 0.826150 0.563451i \(-0.190526\pi\)
0.826150 + 0.563451i \(0.190526\pi\)
\(752\) 0 0
\(753\) −6.12312e24 −2.15881
\(754\) 0 0
\(755\) −2.04500e23 −0.0704927
\(756\) 0 0
\(757\) 3.43112e23 0.115644 0.0578218 0.998327i \(-0.481584\pi\)
0.0578218 + 0.998327i \(0.481584\pi\)
\(758\) 0 0
\(759\) −4.10394e24 −1.35253
\(760\) 0 0
\(761\) 3.01271e24 0.970930 0.485465 0.874256i \(-0.338650\pi\)
0.485465 + 0.874256i \(0.338650\pi\)
\(762\) 0 0
\(763\) −6.29219e24 −1.98309
\(764\) 0 0
\(765\) 7.19212e23 0.221684
\(766\) 0 0
\(767\) −5.84893e24 −1.76326
\(768\) 0 0
\(769\) −1.62418e23 −0.0478918 −0.0239459 0.999713i \(-0.507623\pi\)
−0.0239459 + 0.999713i \(0.507623\pi\)
\(770\) 0 0
\(771\) 6.48643e24 1.87087
\(772\) 0 0
\(773\) −4.43975e24 −1.25266 −0.626329 0.779559i \(-0.715443\pi\)
−0.626329 + 0.779559i \(0.715443\pi\)
\(774\) 0 0
\(775\) 1.41762e24 0.391288
\(776\) 0 0
\(777\) 4.76921e23 0.128786
\(778\) 0 0
\(779\) −9.69883e22 −0.0256242
\(780\) 0 0
\(781\) −6.03488e24 −1.56004
\(782\) 0 0
\(783\) 2.93537e23 0.0742486
\(784\) 0 0
\(785\) 1.88846e24 0.467429
\(786\) 0 0
\(787\) 2.95546e24 0.715880 0.357940 0.933745i \(-0.383479\pi\)
0.357940 + 0.933745i \(0.383479\pi\)
\(788\) 0 0
\(789\) 1.53613e24 0.364145
\(790\) 0 0
\(791\) −2.49705e24 −0.579331
\(792\) 0 0
\(793\) 7.97695e24 1.81140
\(794\) 0 0
\(795\) 2.69382e24 0.598752
\(796\) 0 0
\(797\) −6.01045e24 −1.30771 −0.653855 0.756620i \(-0.726849\pi\)
−0.653855 + 0.756620i \(0.726849\pi\)
\(798\) 0 0
\(799\) −3.30808e24 −0.704577
\(800\) 0 0
\(801\) 3.47563e24 0.724698
\(802\) 0 0
\(803\) −8.35376e24 −1.70530
\(804\) 0 0
\(805\) 1.70576e24 0.340921
\(806\) 0 0
\(807\) 8.17663e24 1.60011
\(808\) 0 0
\(809\) 7.16704e24 1.37334 0.686668 0.726971i \(-0.259072\pi\)
0.686668 + 0.726971i \(0.259072\pi\)
\(810\) 0 0
\(811\) 5.73123e24 1.07540 0.537701 0.843136i \(-0.319293\pi\)
0.537701 + 0.843136i \(0.319293\pi\)
\(812\) 0 0
\(813\) 6.99170e24 1.28473
\(814\) 0 0
\(815\) −3.83171e24 −0.689529
\(816\) 0 0
\(817\) 6.78745e22 0.0119624
\(818\) 0 0
\(819\) −1.00745e25 −1.73905
\(820\) 0 0
\(821\) 2.45653e24 0.415341 0.207670 0.978199i \(-0.433412\pi\)
0.207670 + 0.978199i \(0.433412\pi\)
\(822\) 0 0
\(823\) −1.00636e25 −1.66670 −0.833348 0.552749i \(-0.813579\pi\)
−0.833348 + 0.552749i \(0.813579\pi\)
\(824\) 0 0
\(825\) −1.31468e25 −2.13286
\(826\) 0 0
\(827\) 2.31323e24 0.367639 0.183820 0.982960i \(-0.441154\pi\)
0.183820 + 0.982960i \(0.441154\pi\)
\(828\) 0 0
\(829\) 5.49125e24 0.854983 0.427491 0.904019i \(-0.359397\pi\)
0.427491 + 0.904019i \(0.359397\pi\)
\(830\) 0 0
\(831\) −2.13080e24 −0.325038
\(832\) 0 0
\(833\) −5.60185e24 −0.837237
\(834\) 0 0
\(835\) −4.06016e24 −0.594577
\(836\) 0 0
\(837\) 4.86126e23 0.0697561
\(838\) 0 0
\(839\) 2.39004e24 0.336069 0.168035 0.985781i \(-0.446258\pi\)
0.168035 + 0.985781i \(0.446258\pi\)
\(840\) 0 0
\(841\) −5.40186e24 −0.744351
\(842\) 0 0
\(843\) 1.97628e24 0.266879
\(844\) 0 0
\(845\) −1.80336e24 −0.238671
\(846\) 0 0
\(847\) 3.04533e25 3.95026
\(848\) 0 0
\(849\) −1.53497e24 −0.195158
\(850\) 0 0
\(851\) −2.52370e23 −0.0314511
\(852\) 0 0
\(853\) −7.04660e24 −0.860821 −0.430410 0.902633i \(-0.641631\pi\)
−0.430410 + 0.902633i \(0.641631\pi\)
\(854\) 0 0
\(855\) 1.09630e23 0.0131286
\(856\) 0 0
\(857\) −6.71395e24 −0.788208 −0.394104 0.919066i \(-0.628945\pi\)
−0.394104 + 0.919066i \(0.628945\pi\)
\(858\) 0 0
\(859\) 8.58583e24 0.988189 0.494095 0.869408i \(-0.335500\pi\)
0.494095 + 0.869408i \(0.335500\pi\)
\(860\) 0 0
\(861\) 1.38753e25 1.56572
\(862\) 0 0
\(863\) −5.52514e24 −0.611296 −0.305648 0.952145i \(-0.598873\pi\)
−0.305648 + 0.952145i \(0.598873\pi\)
\(864\) 0 0
\(865\) 5.07214e23 0.0550242
\(866\) 0 0
\(867\) −8.41585e24 −0.895232
\(868\) 0 0
\(869\) −1.06440e24 −0.111030
\(870\) 0 0
\(871\) −5.68364e24 −0.581396
\(872\) 0 0
\(873\) 3.90294e24 0.391536
\(874\) 0 0
\(875\) 1.20981e25 1.19028
\(876\) 0 0
\(877\) −6.54824e24 −0.631871 −0.315935 0.948781i \(-0.602318\pi\)
−0.315935 + 0.948781i \(0.602318\pi\)
\(878\) 0 0
\(879\) 1.58867e25 1.50358
\(880\) 0 0
\(881\) 5.78483e24 0.537026 0.268513 0.963276i \(-0.413468\pi\)
0.268513 + 0.963276i \(0.413468\pi\)
\(882\) 0 0
\(883\) −1.75999e25 −1.60267 −0.801337 0.598213i \(-0.795878\pi\)
−0.801337 + 0.598213i \(0.795878\pi\)
\(884\) 0 0
\(885\) −9.10552e24 −0.813368
\(886\) 0 0
\(887\) 1.06620e25 0.934300 0.467150 0.884178i \(-0.345281\pi\)
0.467150 + 0.884178i \(0.345281\pi\)
\(888\) 0 0
\(889\) −1.87831e25 −1.61474
\(890\) 0 0
\(891\) −2.44389e25 −2.06120
\(892\) 0 0
\(893\) −5.04254e23 −0.0417265
\(894\) 0 0
\(895\) −5.50862e24 −0.447246
\(896\) 0 0
\(897\) 1.12991e25 0.900136
\(898\) 0 0
\(899\) 3.07253e24 0.240181
\(900\) 0 0
\(901\) 7.98634e24 0.612615
\(902\) 0 0
\(903\) −9.71023e24 −0.730943
\(904\) 0 0
\(905\) 1.32310e23 0.00977417
\(906\) 0 0
\(907\) −1.63230e25 −1.18342 −0.591709 0.806152i \(-0.701547\pi\)
−0.591709 + 0.806152i \(0.701547\pi\)
\(908\) 0 0
\(909\) 6.34792e24 0.451688
\(910\) 0 0
\(911\) 1.45971e25 1.01944 0.509719 0.860341i \(-0.329749\pi\)
0.509719 + 0.860341i \(0.329749\pi\)
\(912\) 0 0
\(913\) 3.51456e25 2.40918
\(914\) 0 0
\(915\) 1.24184e25 0.835577
\(916\) 0 0
\(917\) −1.70018e25 −1.12294
\(918\) 0 0
\(919\) 1.83968e24 0.119278 0.0596390 0.998220i \(-0.481005\pi\)
0.0596390 + 0.998220i \(0.481005\pi\)
\(920\) 0 0
\(921\) 1.64019e25 1.04397
\(922\) 0 0
\(923\) 1.66154e25 1.03824
\(924\) 0 0
\(925\) −8.08457e23 −0.0495966
\(926\) 0 0
\(927\) −3.45716e24 −0.208230
\(928\) 0 0
\(929\) −7.16411e24 −0.423671 −0.211835 0.977305i \(-0.567944\pi\)
−0.211835 + 0.977305i \(0.567944\pi\)
\(930\) 0 0
\(931\) −8.53894e23 −0.0495829
\(932\) 0 0
\(933\) −2.71875e25 −1.55016
\(934\) 0 0
\(935\) 8.34108e24 0.467007
\(936\) 0 0
\(937\) −1.68079e25 −0.924119 −0.462059 0.886849i \(-0.652889\pi\)
−0.462059 + 0.886849i \(0.652889\pi\)
\(938\) 0 0
\(939\) −4.00920e25 −2.16471
\(940\) 0 0
\(941\) 2.99139e25 1.58621 0.793105 0.609085i \(-0.208463\pi\)
0.793105 + 0.609085i \(0.208463\pi\)
\(942\) 0 0
\(943\) −7.34232e24 −0.382370
\(944\) 0 0
\(945\) 1.87381e24 0.0958420
\(946\) 0 0
\(947\) 1.00925e25 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(948\) 0 0
\(949\) 2.29998e25 1.13491
\(950\) 0 0
\(951\) −4.79245e24 −0.232286
\(952\) 0 0
\(953\) 9.95941e24 0.474181 0.237090 0.971488i \(-0.423806\pi\)
0.237090 + 0.971488i \(0.423806\pi\)
\(954\) 0 0
\(955\) −3.29288e24 −0.154009
\(956\) 0 0
\(957\) −2.84941e25 −1.30919
\(958\) 0 0
\(959\) −4.61963e25 −2.08521
\(960\) 0 0
\(961\) −1.74617e25 −0.774351
\(962\) 0 0
\(963\) 2.11655e25 0.922158
\(964\) 0 0
\(965\) 1.05539e25 0.451785
\(966\) 0 0
\(967\) −4.62055e24 −0.194343 −0.0971715 0.995268i \(-0.530980\pi\)
−0.0971715 + 0.995268i \(0.530980\pi\)
\(968\) 0 0
\(969\) 6.88872e23 0.0284699
\(970\) 0 0
\(971\) −1.50646e24 −0.0611780 −0.0305890 0.999532i \(-0.509738\pi\)
−0.0305890 + 0.999532i \(0.509738\pi\)
\(972\) 0 0
\(973\) 2.91796e25 1.16445
\(974\) 0 0
\(975\) 3.61962e25 1.41946
\(976\) 0 0
\(977\) −2.16292e25 −0.833560 −0.416780 0.909007i \(-0.636841\pi\)
−0.416780 + 0.909007i \(0.636841\pi\)
\(978\) 0 0
\(979\) 4.03087e25 1.52667
\(980\) 0 0
\(981\) −3.06145e25 −1.13957
\(982\) 0 0
\(983\) 1.09543e25 0.400754 0.200377 0.979719i \(-0.435783\pi\)
0.200377 + 0.979719i \(0.435783\pi\)
\(984\) 0 0
\(985\) −1.94290e25 −0.698622
\(986\) 0 0
\(987\) 7.21393e25 2.54962
\(988\) 0 0
\(989\) 5.13831e24 0.178506
\(990\) 0 0
\(991\) −2.10377e25 −0.718410 −0.359205 0.933259i \(-0.616952\pi\)
−0.359205 + 0.933259i \(0.616952\pi\)
\(992\) 0 0
\(993\) 7.29913e25 2.45020
\(994\) 0 0
\(995\) −9.27549e24 −0.306084
\(996\) 0 0
\(997\) −2.15374e25 −0.698690 −0.349345 0.936994i \(-0.613596\pi\)
−0.349345 + 0.936994i \(0.613596\pi\)
\(998\) 0 0
\(999\) −2.77233e23 −0.00884175
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 4.18.a.a.1.2 2
3.2 odd 2 36.18.a.d.1.2 2
4.3 odd 2 16.18.a.e.1.1 2
5.2 odd 4 100.18.c.a.49.2 4
5.3 odd 4 100.18.c.a.49.3 4
5.4 even 2 100.18.a.b.1.1 2
8.3 odd 2 64.18.a.g.1.2 2
8.5 even 2 64.18.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.18.a.a.1.2 2 1.1 even 1 trivial
16.18.a.e.1.1 2 4.3 odd 2
36.18.a.d.1.2 2 3.2 odd 2
64.18.a.g.1.2 2 8.3 odd 2
64.18.a.l.1.1 2 8.5 even 2
100.18.a.b.1.1 2 5.4 even 2
100.18.c.a.49.2 4 5.2 odd 4
100.18.c.a.49.3 4 5.3 odd 4