Properties

Label 2736.3.m.f.1711.11
Level $2736$
Weight $3$
Character 2736.1711
Analytic conductor $74.551$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,3,Mod(1711,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.1711");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.m (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + 48457 x^{4} - 30372 x^{3} + 89392 x^{2} + 9344 x + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1711.11
Root \(-3.37682 + 5.84883i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1711
Dual form 2736.3.m.f.1711.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.26675 q^{5} -9.51333i q^{7} +O(q^{10})\) \(q+8.26675 q^{5} -9.51333i q^{7} +19.9729i q^{11} +7.11018 q^{13} +19.2396 q^{17} +4.35890i q^{19} +28.6988i q^{23} +43.3391 q^{25} +6.08981 q^{29} +0.434440i q^{31} -78.6443i q^{35} +3.56772 q^{37} -52.7023 q^{41} +67.7920i q^{43} +16.7325i q^{47} -41.5034 q^{49} +93.2965 q^{53} +165.111i q^{55} +92.1130i q^{59} -91.4439 q^{61} +58.7780 q^{65} +69.1041i q^{67} -111.081i q^{71} -11.0069 q^{73} +190.009 q^{77} +40.2810i q^{79} -25.1933i q^{83} +159.049 q^{85} -156.317 q^{89} -67.6414i q^{91} +36.0339i q^{95} +177.845 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{5} + 36 q^{13} + 10 q^{17} + 58 q^{25} - 12 q^{29} + 32 q^{37} - 136 q^{41} - 22 q^{49} + 236 q^{53} - 210 q^{61} + 52 q^{65} - 158 q^{73} + 70 q^{77} + 242 q^{85} - 444 q^{89} + 320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 8.26675 1.65335 0.826675 0.562680i \(-0.190230\pi\)
0.826675 + 0.562680i \(0.190230\pi\)
\(6\) 0 0
\(7\) − 9.51333i − 1.35905i −0.733654 0.679523i \(-0.762187\pi\)
0.733654 0.679523i \(-0.237813\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 19.9729i 1.81572i 0.419272 + 0.907861i \(0.362286\pi\)
−0.419272 + 0.907861i \(0.637714\pi\)
\(12\) 0 0
\(13\) 7.11018 0.546937 0.273468 0.961881i \(-0.411829\pi\)
0.273468 + 0.961881i \(0.411829\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 19.2396 1.13174 0.565870 0.824495i \(-0.308540\pi\)
0.565870 + 0.824495i \(0.308540\pi\)
\(18\) 0 0
\(19\) 4.35890i 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 28.6988i 1.24778i 0.781514 + 0.623888i \(0.214448\pi\)
−0.781514 + 0.623888i \(0.785552\pi\)
\(24\) 0 0
\(25\) 43.3391 1.73356
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.08981 0.209993 0.104997 0.994473i \(-0.466517\pi\)
0.104997 + 0.994473i \(0.466517\pi\)
\(30\) 0 0
\(31\) 0.434440i 0.0140142i 0.999975 + 0.00700709i \(0.00223044\pi\)
−0.999975 + 0.00700709i \(0.997770\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 78.6443i − 2.24698i
\(36\) 0 0
\(37\) 3.56772 0.0964248 0.0482124 0.998837i \(-0.484648\pi\)
0.0482124 + 0.998837i \(0.484648\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −52.7023 −1.28542 −0.642711 0.766108i \(-0.722191\pi\)
−0.642711 + 0.766108i \(0.722191\pi\)
\(42\) 0 0
\(43\) 67.7920i 1.57656i 0.615318 + 0.788279i \(0.289028\pi\)
−0.615318 + 0.788279i \(0.710972\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.7325i 0.356011i 0.984029 + 0.178006i \(0.0569645\pi\)
−0.984029 + 0.178006i \(0.943035\pi\)
\(48\) 0 0
\(49\) −41.5034 −0.847008
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 93.2965 1.76031 0.880156 0.474685i \(-0.157438\pi\)
0.880156 + 0.474685i \(0.157438\pi\)
\(54\) 0 0
\(55\) 165.111i 3.00202i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 92.1130i 1.56124i 0.625007 + 0.780619i \(0.285096\pi\)
−0.625007 + 0.780619i \(0.714904\pi\)
\(60\) 0 0
\(61\) −91.4439 −1.49908 −0.749540 0.661959i \(-0.769725\pi\)
−0.749540 + 0.661959i \(0.769725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 58.7780 0.904278
\(66\) 0 0
\(67\) 69.1041i 1.03140i 0.856768 + 0.515702i \(0.172469\pi\)
−0.856768 + 0.515702i \(0.827531\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 111.081i − 1.56452i −0.622950 0.782262i \(-0.714066\pi\)
0.622950 0.782262i \(-0.285934\pi\)
\(72\) 0 0
\(73\) −11.0069 −0.150780 −0.0753898 0.997154i \(-0.524020\pi\)
−0.0753898 + 0.997154i \(0.524020\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 190.009 2.46765
\(78\) 0 0
\(79\) 40.2810i 0.509886i 0.966956 + 0.254943i \(0.0820566\pi\)
−0.966956 + 0.254943i \(0.917943\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 25.1933i − 0.303534i −0.988416 0.151767i \(-0.951504\pi\)
0.988416 0.151767i \(-0.0484963\pi\)
\(84\) 0 0
\(85\) 159.049 1.87116
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −156.317 −1.75637 −0.878183 0.478325i \(-0.841244\pi\)
−0.878183 + 0.478325i \(0.841244\pi\)
\(90\) 0 0
\(91\) − 67.6414i − 0.743313i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 36.0339i 0.379304i
\(96\) 0 0
\(97\) 177.845 1.83346 0.916729 0.399509i \(-0.130819\pi\)
0.916729 + 0.399509i \(0.130819\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 139.570 1.38188 0.690942 0.722910i \(-0.257196\pi\)
0.690942 + 0.722910i \(0.257196\pi\)
\(102\) 0 0
\(103\) − 93.8246i − 0.910918i −0.890257 0.455459i \(-0.849475\pi\)
0.890257 0.455459i \(-0.150525\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 62.9124i − 0.587966i −0.955811 0.293983i \(-0.905019\pi\)
0.955811 0.293983i \(-0.0949809\pi\)
\(108\) 0 0
\(109\) −8.14412 −0.0747167 −0.0373583 0.999302i \(-0.511894\pi\)
−0.0373583 + 0.999302i \(0.511894\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −23.6334 −0.209145 −0.104573 0.994517i \(-0.533347\pi\)
−0.104573 + 0.994517i \(0.533347\pi\)
\(114\) 0 0
\(115\) 237.246i 2.06301i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 183.032i − 1.53809i
\(120\) 0 0
\(121\) −277.918 −2.29685
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 151.605 1.21284
\(126\) 0 0
\(127\) 38.4328i 0.302620i 0.988486 + 0.151310i \(0.0483492\pi\)
−0.988486 + 0.151310i \(0.951651\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 78.6785i 0.600599i 0.953845 + 0.300299i \(0.0970866\pi\)
−0.953845 + 0.300299i \(0.902913\pi\)
\(132\) 0 0
\(133\) 41.4676 0.311787
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −147.488 −1.07656 −0.538278 0.842767i \(-0.680925\pi\)
−0.538278 + 0.842767i \(0.680925\pi\)
\(138\) 0 0
\(139\) − 7.30345i − 0.0525428i −0.999655 0.0262714i \(-0.991637\pi\)
0.999655 0.0262714i \(-0.00836341\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 142.011i 0.993085i
\(144\) 0 0
\(145\) 50.3429 0.347192
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.67149 0.0112181 0.00560903 0.999984i \(-0.498215\pi\)
0.00560903 + 0.999984i \(0.498215\pi\)
\(150\) 0 0
\(151\) − 286.836i − 1.89957i −0.312899 0.949786i \(-0.601300\pi\)
0.312899 0.949786i \(-0.398700\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.59140i 0.0231703i
\(156\) 0 0
\(157\) 30.5840 0.194803 0.0974013 0.995245i \(-0.468947\pi\)
0.0974013 + 0.995245i \(0.468947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 273.021 1.69579
\(162\) 0 0
\(163\) − 87.0132i − 0.533823i −0.963721 0.266912i \(-0.913997\pi\)
0.963721 0.266912i \(-0.0860032\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 25.3317i 0.151687i 0.997120 + 0.0758433i \(0.0241649\pi\)
−0.997120 + 0.0758433i \(0.975835\pi\)
\(168\) 0 0
\(169\) −118.445 −0.700860
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 47.0106 0.271738 0.135869 0.990727i \(-0.456617\pi\)
0.135869 + 0.990727i \(0.456617\pi\)
\(174\) 0 0
\(175\) − 412.299i − 2.35599i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.5579i 0.0869159i 0.999055 + 0.0434579i \(0.0138375\pi\)
−0.999055 + 0.0434579i \(0.986163\pi\)
\(180\) 0 0
\(181\) 298.993 1.65189 0.825947 0.563748i \(-0.190641\pi\)
0.825947 + 0.563748i \(0.190641\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 29.4934 0.159424
\(186\) 0 0
\(187\) 384.271i 2.05492i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 317.098i − 1.66020i −0.557614 0.830100i \(-0.688283\pi\)
0.557614 0.830100i \(-0.311717\pi\)
\(192\) 0 0
\(193\) 176.434 0.914168 0.457084 0.889424i \(-0.348894\pi\)
0.457084 + 0.889424i \(0.348894\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 173.653 0.881487 0.440743 0.897633i \(-0.354715\pi\)
0.440743 + 0.897633i \(0.354715\pi\)
\(198\) 0 0
\(199\) − 167.157i − 0.839984i −0.907528 0.419992i \(-0.862033\pi\)
0.907528 0.419992i \(-0.137967\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 57.9343i − 0.285391i
\(204\) 0 0
\(205\) −435.677 −2.12525
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −87.0600 −0.416555
\(210\) 0 0
\(211\) 86.0298i 0.407724i 0.979000 + 0.203862i \(0.0653494\pi\)
−0.979000 + 0.203862i \(0.934651\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 560.419i 2.60660i
\(216\) 0 0
\(217\) 4.13297 0.0190459
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 136.797 0.618990
\(222\) 0 0
\(223\) − 321.231i − 1.44050i −0.693717 0.720248i \(-0.744028\pi\)
0.693717 0.720248i \(-0.255972\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 204.788i − 0.902152i −0.892486 0.451076i \(-0.851040\pi\)
0.892486 0.451076i \(-0.148960\pi\)
\(228\) 0 0
\(229\) 31.7066 0.138457 0.0692283 0.997601i \(-0.477946\pi\)
0.0692283 + 0.997601i \(0.477946\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 351.865 1.51015 0.755074 0.655640i \(-0.227601\pi\)
0.755074 + 0.655640i \(0.227601\pi\)
\(234\) 0 0
\(235\) 138.324i 0.588611i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 287.952i 1.20482i 0.798188 + 0.602409i \(0.205792\pi\)
−0.798188 + 0.602409i \(0.794208\pi\)
\(240\) 0 0
\(241\) −341.159 −1.41560 −0.707799 0.706414i \(-0.750312\pi\)
−0.707799 + 0.706414i \(0.750312\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −343.098 −1.40040
\(246\) 0 0
\(247\) 30.9925i 0.125476i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 83.6498i 0.333266i 0.986019 + 0.166633i \(0.0532896\pi\)
−0.986019 + 0.166633i \(0.946710\pi\)
\(252\) 0 0
\(253\) −573.200 −2.26561
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −256.627 −0.998547 −0.499274 0.866444i \(-0.666400\pi\)
−0.499274 + 0.866444i \(0.666400\pi\)
\(258\) 0 0
\(259\) − 33.9408i − 0.131046i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 35.0162i − 0.133141i −0.997782 0.0665707i \(-0.978794\pi\)
0.997782 0.0665707i \(-0.0212058\pi\)
\(264\) 0 0
\(265\) 771.259 2.91041
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 242.681 0.902161 0.451081 0.892483i \(-0.351039\pi\)
0.451081 + 0.892483i \(0.351039\pi\)
\(270\) 0 0
\(271\) 319.878i 1.18036i 0.807270 + 0.590182i \(0.200944\pi\)
−0.807270 + 0.590182i \(0.799056\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 865.609i 3.14767i
\(276\) 0 0
\(277\) −185.014 −0.667921 −0.333960 0.942587i \(-0.608385\pi\)
−0.333960 + 0.942587i \(0.608385\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 258.307 0.919243 0.459621 0.888115i \(-0.347985\pi\)
0.459621 + 0.888115i \(0.347985\pi\)
\(282\) 0 0
\(283\) − 210.101i − 0.742408i −0.928551 0.371204i \(-0.878945\pi\)
0.928551 0.371204i \(-0.121055\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 501.375i 1.74695i
\(288\) 0 0
\(289\) 81.1612 0.280835
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 327.177 1.11665 0.558323 0.829624i \(-0.311445\pi\)
0.558323 + 0.829624i \(0.311445\pi\)
\(294\) 0 0
\(295\) 761.475i 2.58127i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 204.054i 0.682454i
\(300\) 0 0
\(301\) 644.927 2.14262
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −755.943 −2.47850
\(306\) 0 0
\(307\) 65.9862i 0.214939i 0.994208 + 0.107469i \(0.0342748\pi\)
−0.994208 + 0.107469i \(0.965725\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 241.984i 0.778084i 0.921220 + 0.389042i \(0.127194\pi\)
−0.921220 + 0.389042i \(0.872806\pi\)
\(312\) 0 0
\(313\) 384.453 1.22828 0.614142 0.789195i \(-0.289502\pi\)
0.614142 + 0.789195i \(0.289502\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −524.546 −1.65472 −0.827359 0.561673i \(-0.810158\pi\)
−0.827359 + 0.561673i \(0.810158\pi\)
\(318\) 0 0
\(319\) 121.631i 0.381289i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 83.8634i 0.259639i
\(324\) 0 0
\(325\) 308.149 0.948150
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 159.182 0.483836
\(330\) 0 0
\(331\) − 286.540i − 0.865681i −0.901471 0.432840i \(-0.857511\pi\)
0.901471 0.432840i \(-0.142489\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 571.266i 1.70527i
\(336\) 0 0
\(337\) 418.094 1.24063 0.620317 0.784351i \(-0.287004\pi\)
0.620317 + 0.784351i \(0.287004\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.67703 −0.0254458
\(342\) 0 0
\(343\) − 71.3179i − 0.207924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 296.826i 0.855407i 0.903919 + 0.427703i \(0.140677\pi\)
−0.903919 + 0.427703i \(0.859323\pi\)
\(348\) 0 0
\(349\) 65.1648 0.186719 0.0933593 0.995632i \(-0.470239\pi\)
0.0933593 + 0.995632i \(0.470239\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 334.322 0.947087 0.473544 0.880770i \(-0.342975\pi\)
0.473544 + 0.880770i \(0.342975\pi\)
\(354\) 0 0
\(355\) − 918.280i − 2.58670i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 100.554i − 0.280094i −0.990145 0.140047i \(-0.955275\pi\)
0.990145 0.140047i \(-0.0447253\pi\)
\(360\) 0 0
\(361\) −19.0000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −90.9914 −0.249291
\(366\) 0 0
\(367\) 247.124i 0.673362i 0.941619 + 0.336681i \(0.109304\pi\)
−0.941619 + 0.336681i \(0.890696\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 887.560i − 2.39235i
\(372\) 0 0
\(373\) 301.594 0.808563 0.404282 0.914635i \(-0.367522\pi\)
0.404282 + 0.914635i \(0.367522\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 43.2996 0.114853
\(378\) 0 0
\(379\) 17.0620i 0.0450186i 0.999747 + 0.0225093i \(0.00716553\pi\)
−0.999747 + 0.0225093i \(0.992834\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 505.600i − 1.32010i −0.751220 0.660052i \(-0.770534\pi\)
0.751220 0.660052i \(-0.229466\pi\)
\(384\) 0 0
\(385\) 1570.76 4.07989
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −62.8932 −0.161679 −0.0808396 0.996727i \(-0.525760\pi\)
−0.0808396 + 0.996727i \(0.525760\pi\)
\(390\) 0 0
\(391\) 552.153i 1.41216i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 332.993i 0.843019i
\(396\) 0 0
\(397\) −179.718 −0.452690 −0.226345 0.974047i \(-0.572678\pi\)
−0.226345 + 0.974047i \(0.572678\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 545.788 1.36107 0.680534 0.732716i \(-0.261748\pi\)
0.680534 + 0.732716i \(0.261748\pi\)
\(402\) 0 0
\(403\) 3.08894i 0.00766487i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 71.2578i 0.175081i
\(408\) 0 0
\(409\) 437.863 1.07057 0.535285 0.844671i \(-0.320204\pi\)
0.535285 + 0.844671i \(0.320204\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 876.301 2.12179
\(414\) 0 0
\(415\) − 208.267i − 0.501848i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 291.925i 0.696719i 0.937361 + 0.348359i \(0.113261\pi\)
−0.937361 + 0.348359i \(0.886739\pi\)
\(420\) 0 0
\(421\) −372.544 −0.884902 −0.442451 0.896793i \(-0.645891\pi\)
−0.442451 + 0.896793i \(0.645891\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 833.826 1.96194
\(426\) 0 0
\(427\) 869.935i 2.03732i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 13.6862i − 0.0317546i −0.999874 0.0158773i \(-0.994946\pi\)
0.999874 0.0158773i \(-0.00505411\pi\)
\(432\) 0 0
\(433\) −830.034 −1.91694 −0.958469 0.285197i \(-0.907941\pi\)
−0.958469 + 0.285197i \(0.907941\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −125.095 −0.286259
\(438\) 0 0
\(439\) − 753.055i − 1.71539i −0.514161 0.857694i \(-0.671897\pi\)
0.514161 0.857694i \(-0.328103\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 320.555i − 0.723600i −0.932256 0.361800i \(-0.882162\pi\)
0.932256 0.361800i \(-0.117838\pi\)
\(444\) 0 0
\(445\) −1292.23 −2.90389
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 602.223 1.34125 0.670627 0.741795i \(-0.266025\pi\)
0.670627 + 0.741795i \(0.266025\pi\)
\(450\) 0 0
\(451\) − 1052.62i − 2.33397i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 559.175i − 1.22896i
\(456\) 0 0
\(457\) −450.930 −0.986719 −0.493359 0.869826i \(-0.664231\pi\)
−0.493359 + 0.869826i \(0.664231\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −83.5485 −0.181233 −0.0906166 0.995886i \(-0.528884\pi\)
−0.0906166 + 0.995886i \(0.528884\pi\)
\(462\) 0 0
\(463\) 167.556i 0.361893i 0.983493 + 0.180946i \(0.0579161\pi\)
−0.983493 + 0.180946i \(0.942084\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 448.878i 0.961196i 0.876941 + 0.480598i \(0.159580\pi\)
−0.876941 + 0.480598i \(0.840420\pi\)
\(468\) 0 0
\(469\) 657.410 1.40173
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1354.01 −2.86259
\(474\) 0 0
\(475\) 188.911i 0.397707i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 269.049i − 0.561690i −0.959753 0.280845i \(-0.909385\pi\)
0.959753 0.280845i \(-0.0906146\pi\)
\(480\) 0 0
\(481\) 25.3671 0.0527383
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1470.20 3.03135
\(486\) 0 0
\(487\) 398.113i 0.817481i 0.912651 + 0.408741i \(0.134032\pi\)
−0.912651 + 0.408741i \(0.865968\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 638.967i − 1.30136i −0.759353 0.650679i \(-0.774484\pi\)
0.759353 0.650679i \(-0.225516\pi\)
\(492\) 0 0
\(493\) 117.165 0.237658
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1056.75 −2.12626
\(498\) 0 0
\(499\) − 289.639i − 0.580439i −0.956960 0.290220i \(-0.906272\pi\)
0.956960 0.290220i \(-0.0937283\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 635.303i − 1.26303i −0.775365 0.631514i \(-0.782434\pi\)
0.775365 0.631514i \(-0.217566\pi\)
\(504\) 0 0
\(505\) 1153.79 2.28474
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 547.921 1.07647 0.538233 0.842796i \(-0.319092\pi\)
0.538233 + 0.842796i \(0.319092\pi\)
\(510\) 0 0
\(511\) 104.712i 0.204917i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 775.624i − 1.50607i
\(516\) 0 0
\(517\) −334.198 −0.646417
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −127.108 −0.243969 −0.121985 0.992532i \(-0.538926\pi\)
−0.121985 + 0.992532i \(0.538926\pi\)
\(522\) 0 0
\(523\) 431.535i 0.825115i 0.910932 + 0.412557i \(0.135364\pi\)
−0.910932 + 0.412557i \(0.864636\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.35843i 0.0158604i
\(528\) 0 0
\(529\) −294.623 −0.556944
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −374.723 −0.703045
\(534\) 0 0
\(535\) − 520.081i − 0.972113i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 828.944i − 1.53793i
\(540\) 0 0
\(541\) −999.802 −1.84806 −0.924032 0.382316i \(-0.875127\pi\)
−0.924032 + 0.382316i \(0.875127\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −67.3254 −0.123533
\(546\) 0 0
\(547\) 236.004i 0.431451i 0.976454 + 0.215726i \(0.0692117\pi\)
−0.976454 + 0.215726i \(0.930788\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 26.5449i 0.0481758i
\(552\) 0 0
\(553\) 383.206 0.692958
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 866.326 1.55534 0.777671 0.628671i \(-0.216401\pi\)
0.777671 + 0.628671i \(0.216401\pi\)
\(558\) 0 0
\(559\) 482.013i 0.862278i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 587.885i − 1.04420i −0.852884 0.522101i \(-0.825149\pi\)
0.852884 0.522101i \(-0.174851\pi\)
\(564\) 0 0
\(565\) −195.371 −0.345790
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −191.363 −0.336314 −0.168157 0.985760i \(-0.553782\pi\)
−0.168157 + 0.985760i \(0.553782\pi\)
\(570\) 0 0
\(571\) − 287.222i − 0.503016i −0.967855 0.251508i \(-0.919073\pi\)
0.967855 0.251508i \(-0.0809266\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1243.78i 2.16310i
\(576\) 0 0
\(577\) −443.562 −0.768738 −0.384369 0.923180i \(-0.625581\pi\)
−0.384369 + 0.923180i \(0.625581\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −239.672 −0.412517
\(582\) 0 0
\(583\) 1863.41i 3.19624i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 514.692i − 0.876817i −0.898776 0.438408i \(-0.855542\pi\)
0.898776 0.438408i \(-0.144458\pi\)
\(588\) 0 0
\(589\) −1.89368 −0.00321507
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −227.211 −0.383156 −0.191578 0.981477i \(-0.561360\pi\)
−0.191578 + 0.981477i \(0.561360\pi\)
\(594\) 0 0
\(595\) − 1513.08i − 2.54300i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 805.147i 1.34415i 0.740482 + 0.672076i \(0.234597\pi\)
−0.740482 + 0.672076i \(0.765403\pi\)
\(600\) 0 0
\(601\) −1110.23 −1.84730 −0.923649 0.383238i \(-0.874809\pi\)
−0.923649 + 0.383238i \(0.874809\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2297.48 −3.79749
\(606\) 0 0
\(607\) 293.733i 0.483909i 0.970288 + 0.241955i \(0.0777886\pi\)
−0.970288 + 0.241955i \(0.922211\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 118.971i 0.194716i
\(612\) 0 0
\(613\) 702.779 1.14646 0.573230 0.819395i \(-0.305690\pi\)
0.573230 + 0.819395i \(0.305690\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 823.243 1.33427 0.667134 0.744938i \(-0.267521\pi\)
0.667134 + 0.744938i \(0.267521\pi\)
\(618\) 0 0
\(619\) 400.632i 0.647225i 0.946190 + 0.323612i \(0.104897\pi\)
−0.946190 + 0.323612i \(0.895103\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1487.09i 2.38698i
\(624\) 0 0
\(625\) 169.800 0.271680
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 68.6413 0.109128
\(630\) 0 0
\(631\) − 354.013i − 0.561034i −0.959849 0.280517i \(-0.909494\pi\)
0.959849 0.280517i \(-0.0905059\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 317.714i 0.500337i
\(636\) 0 0
\(637\) −295.096 −0.463260
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −577.597 −0.901088 −0.450544 0.892754i \(-0.648770\pi\)
−0.450544 + 0.892754i \(0.648770\pi\)
\(642\) 0 0
\(643\) 791.746i 1.23133i 0.788007 + 0.615666i \(0.211113\pi\)
−0.788007 + 0.615666i \(0.788887\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1047.43i 1.61890i 0.587192 + 0.809448i \(0.300234\pi\)
−0.587192 + 0.809448i \(0.699766\pi\)
\(648\) 0 0
\(649\) −1839.77 −2.83477
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −815.274 −1.24851 −0.624253 0.781222i \(-0.714596\pi\)
−0.624253 + 0.781222i \(0.714596\pi\)
\(654\) 0 0
\(655\) 650.415i 0.993000i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 238.992i − 0.362659i −0.983422 0.181330i \(-0.941960\pi\)
0.983422 0.181330i \(-0.0580401\pi\)
\(660\) 0 0
\(661\) −542.358 −0.820511 −0.410256 0.911971i \(-0.634561\pi\)
−0.410256 + 0.911971i \(0.634561\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 342.802 0.515492
\(666\) 0 0
\(667\) 174.770i 0.262025i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 1826.40i − 2.72191i
\(672\) 0 0
\(673\) 468.193 0.695681 0.347840 0.937554i \(-0.386915\pi\)
0.347840 + 0.937554i \(0.386915\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −771.400 −1.13944 −0.569719 0.821839i \(-0.692948\pi\)
−0.569719 + 0.821839i \(0.692948\pi\)
\(678\) 0 0
\(679\) − 1691.90i − 2.49176i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 821.359i − 1.20258i −0.799033 0.601288i \(-0.794655\pi\)
0.799033 0.601288i \(-0.205345\pi\)
\(684\) 0 0
\(685\) −1219.25 −1.77992
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 663.355 0.962779
\(690\) 0 0
\(691\) − 971.147i − 1.40542i −0.711475 0.702711i \(-0.751972\pi\)
0.711475 0.702711i \(-0.248028\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 60.3758i − 0.0868716i
\(696\) 0 0
\(697\) −1013.97 −1.45476
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 128.066 0.182691 0.0913453 0.995819i \(-0.470883\pi\)
0.0913453 + 0.995819i \(0.470883\pi\)
\(702\) 0 0
\(703\) 15.5513i 0.0221214i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1327.78i − 1.87804i
\(708\) 0 0
\(709\) −44.7470 −0.0631128 −0.0315564 0.999502i \(-0.510046\pi\)
−0.0315564 + 0.999502i \(0.510046\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.4679 −0.0174866
\(714\) 0 0
\(715\) 1173.97i 1.64192i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 298.311i − 0.414897i −0.978246 0.207448i \(-0.933484\pi\)
0.978246 0.207448i \(-0.0665159\pi\)
\(720\) 0 0
\(721\) −892.584 −1.23798
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 263.927 0.364037
\(726\) 0 0
\(727\) − 1306.36i − 1.79692i −0.439052 0.898461i \(-0.644686\pi\)
0.439052 0.898461i \(-0.355314\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1304.29i 1.78425i
\(732\) 0 0
\(733\) −480.550 −0.655593 −0.327796 0.944748i \(-0.606306\pi\)
−0.327796 + 0.944748i \(0.606306\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1380.21 −1.87274
\(738\) 0 0
\(739\) 748.985i 1.01351i 0.862090 + 0.506756i \(0.169155\pi\)
−0.862090 + 0.506756i \(0.830845\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 397.466i − 0.534948i −0.963565 0.267474i \(-0.913811\pi\)
0.963565 0.267474i \(-0.0861890\pi\)
\(744\) 0 0
\(745\) 13.8178 0.0185474
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −598.506 −0.799073
\(750\) 0 0
\(751\) 1287.95i 1.71498i 0.514504 + 0.857488i \(0.327976\pi\)
−0.514504 + 0.857488i \(0.672024\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 2371.20i − 3.14066i
\(756\) 0 0
\(757\) −812.408 −1.07319 −0.536597 0.843838i \(-0.680291\pi\)
−0.536597 + 0.843838i \(0.680291\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 680.285 0.893935 0.446968 0.894550i \(-0.352504\pi\)
0.446968 + 0.894550i \(0.352504\pi\)
\(762\) 0 0
\(763\) 77.4776i 0.101543i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 654.940i 0.853898i
\(768\) 0 0
\(769\) −61.3875 −0.0798277 −0.0399139 0.999203i \(-0.512708\pi\)
−0.0399139 + 0.999203i \(0.512708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −597.488 −0.772947 −0.386473 0.922301i \(-0.626307\pi\)
−0.386473 + 0.922301i \(0.626307\pi\)
\(774\) 0 0
\(775\) 18.8282i 0.0242945i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 229.724i − 0.294896i
\(780\) 0 0
\(781\) 2218.62 2.84074
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 252.830 0.322077
\(786\) 0 0
\(787\) − 1493.71i − 1.89798i −0.315313 0.948988i \(-0.602109\pi\)
0.315313 0.948988i \(-0.397891\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 224.832i 0.284238i
\(792\) 0 0
\(793\) −650.182 −0.819902
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 95.1729 0.119414 0.0597070 0.998216i \(-0.480983\pi\)
0.0597070 + 0.998216i \(0.480983\pi\)
\(798\) 0 0
\(799\) 321.927i 0.402912i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 219.840i − 0.273774i
\(804\) 0 0
\(805\) 2257.00 2.80373
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −916.775 −1.13322 −0.566610 0.823986i \(-0.691745\pi\)
−0.566610 + 0.823986i \(0.691745\pi\)
\(810\) 0 0
\(811\) − 577.775i − 0.712423i −0.934405 0.356211i \(-0.884068\pi\)
0.934405 0.356211i \(-0.115932\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 719.316i − 0.882596i
\(816\) 0 0
\(817\) −295.498 −0.361687
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 854.788 1.04115 0.520577 0.853815i \(-0.325717\pi\)
0.520577 + 0.853815i \(0.325717\pi\)
\(822\) 0 0
\(823\) 963.867i 1.17116i 0.810614 + 0.585581i \(0.199134\pi\)
−0.810614 + 0.585581i \(0.800866\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 476.984i 0.576764i 0.957515 + 0.288382i \(0.0931172\pi\)
−0.957515 + 0.288382i \(0.906883\pi\)
\(828\) 0 0
\(829\) −30.7605 −0.0371055 −0.0185528 0.999828i \(-0.505906\pi\)
−0.0185528 + 0.999828i \(0.505906\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −798.507 −0.958592
\(834\) 0 0
\(835\) 209.411i 0.250791i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 811.970i 0.967783i 0.875128 + 0.483892i \(0.160777\pi\)
−0.875128 + 0.483892i \(0.839223\pi\)
\(840\) 0 0
\(841\) −803.914 −0.955903
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −979.158 −1.15877
\(846\) 0 0
\(847\) 2643.93i 3.12152i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 102.389i 0.120316i
\(852\) 0 0
\(853\) 122.095 0.143136 0.0715682 0.997436i \(-0.477200\pi\)
0.0715682 + 0.997436i \(0.477200\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −577.474 −0.673832 −0.336916 0.941535i \(-0.609384\pi\)
−0.336916 + 0.941535i \(0.609384\pi\)
\(858\) 0 0
\(859\) − 1651.98i − 1.92314i −0.274556 0.961571i \(-0.588531\pi\)
0.274556 0.961571i \(-0.411469\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 479.262i − 0.555344i −0.960676 0.277672i \(-0.910437\pi\)
0.960676 0.277672i \(-0.0895629\pi\)
\(864\) 0 0
\(865\) 388.625 0.449277
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −804.529 −0.925810
\(870\) 0 0
\(871\) 491.342i 0.564113i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 1442.27i − 1.64830i
\(876\) 0 0
\(877\) 634.569 0.723568 0.361784 0.932262i \(-0.382168\pi\)
0.361784 + 0.932262i \(0.382168\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1027.14 −1.16588 −0.582939 0.812516i \(-0.698097\pi\)
−0.582939 + 0.812516i \(0.698097\pi\)
\(882\) 0 0
\(883\) − 539.788i − 0.611312i −0.952142 0.305656i \(-0.901124\pi\)
0.952142 0.305656i \(-0.0988757\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 707.980i − 0.798174i −0.916913 0.399087i \(-0.869327\pi\)
0.916913 0.399087i \(-0.130673\pi\)
\(888\) 0 0
\(889\) 365.624 0.411275
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −72.9354 −0.0816746
\(894\) 0 0
\(895\) 128.614i 0.143702i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.64565i 0.00294288i
\(900\) 0 0
\(901\) 1794.99 1.99221
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2471.70 2.73116
\(906\) 0 0
\(907\) 683.854i 0.753974i 0.926219 + 0.376987i \(0.123040\pi\)
−0.926219 + 0.376987i \(0.876960\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 912.906i − 1.00209i −0.865421 0.501046i \(-0.832949\pi\)
0.865421 0.501046i \(-0.167051\pi\)
\(912\) 0 0
\(913\) 503.185 0.551133
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 748.494 0.816242
\(918\) 0 0
\(919\) − 1549.13i − 1.68567i −0.538175 0.842833i \(-0.680886\pi\)
0.538175 0.842833i \(-0.319114\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 789.807i − 0.855695i
\(924\) 0 0
\(925\) 154.622 0.167159
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1359.01 −1.46287 −0.731436 0.681910i \(-0.761150\pi\)
−0.731436 + 0.681910i \(0.761150\pi\)
\(930\) 0 0
\(931\) − 180.909i − 0.194317i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3176.67i 3.39751i
\(936\) 0 0
\(937\) −1426.90 −1.52284 −0.761420 0.648259i \(-0.775497\pi\)
−0.761420 + 0.648259i \(0.775497\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 922.895 0.980759 0.490380 0.871509i \(-0.336858\pi\)
0.490380 + 0.871509i \(0.336858\pi\)
\(942\) 0 0
\(943\) − 1512.50i − 1.60392i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 416.165i 0.439456i 0.975561 + 0.219728i \(0.0705170\pi\)
−0.975561 + 0.219728i \(0.929483\pi\)
\(948\) 0 0
\(949\) −78.2611 −0.0824669
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1181.47 1.23974 0.619870 0.784705i \(-0.287185\pi\)
0.619870 + 0.784705i \(0.287185\pi\)
\(954\) 0 0
\(955\) − 2621.37i − 2.74489i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1403.10i 1.46309i
\(960\) 0 0
\(961\) 960.811 0.999804
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1458.54 1.51144
\(966\) 0 0
\(967\) 913.101i 0.944262i 0.881528 + 0.472131i \(0.156515\pi\)
−0.881528 + 0.472131i \(0.843485\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 877.256i − 0.903456i −0.892156 0.451728i \(-0.850808\pi\)
0.892156 0.451728i \(-0.149192\pi\)
\(972\) 0 0
\(973\) −69.4801 −0.0714081
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 745.908 0.763467 0.381734 0.924272i \(-0.375327\pi\)
0.381734 + 0.924272i \(0.375327\pi\)
\(978\) 0 0
\(979\) − 3122.10i − 3.18907i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 51.0071i − 0.0518893i −0.999663 0.0259446i \(-0.991741\pi\)
0.999663 0.0259446i \(-0.00825936\pi\)
\(984\) 0 0
\(985\) 1435.54 1.45741
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1945.55 −1.96719
\(990\) 0 0
\(991\) − 87.7408i − 0.0885376i −0.999020 0.0442688i \(-0.985904\pi\)
0.999020 0.0442688i \(-0.0140958\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1381.84i − 1.38879i
\(996\) 0 0
\(997\) 562.992 0.564686 0.282343 0.959314i \(-0.408888\pi\)
0.282343 + 0.959314i \(0.408888\pi\)
\(998\) 0 0
\(999\) 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 2736.3.m.f.1711.11 12
3.2 odd 2 912.3.m.a.799.1 12
4.3 odd 2 inner 2736.3.m.f.1711.12 12
12.11 even 2 912.3.m.a.799.7 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.3.m.a.799.1 12 3.2 odd 2
912.3.m.a.799.7 yes 12 12.11 even 2
2736.3.m.f.1711.11 12 1.1 even 1 trivial
2736.3.m.f.1711.12 12 4.3 odd 2 inner