Properties

Label 324.3.k.a.17.4
Level $324$
Weight $3$
Character 324.17
Analytic conductor $8.828$
Analytic rank $0$
Dimension $36$
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] = [324,3,Mod(17,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.k (of order \(18\), degree \(6\), not 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: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 17.4
Character \(\chi\) \(=\) 324.17
Dual form 324.3.k.a.305.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.32861 + 0.763252i) q^{5} +(-2.73772 + 2.29722i) q^{7} +O(q^{10})\) \(q+(4.32861 + 0.763252i) q^{5} +(-2.73772 + 2.29722i) q^{7} +(14.4582 - 2.54938i) q^{11} +(-3.67778 + 1.33860i) q^{13} +(5.96626 + 3.44462i) q^{17} +(14.1412 + 24.4932i) q^{19} +(-0.832583 + 0.992234i) q^{23} +(-5.33796 - 1.94286i) q^{25} +(12.9959 - 35.7059i) q^{29} +(41.7175 + 35.0052i) q^{31} +(-13.6039 + 7.85422i) q^{35} +(18.5334 - 32.1007i) q^{37} +(14.0768 + 38.6757i) q^{41} +(-0.615899 - 3.49294i) q^{43} +(27.5222 + 32.7997i) q^{47} +(-6.29086 + 35.6773i) q^{49} -47.8007i q^{53} +64.5299 q^{55} +(61.1091 + 10.7752i) q^{59} +(-8.50896 + 7.13987i) q^{61} +(-16.9414 + 2.98722i) q^{65} +(-105.153 + 38.2727i) q^{67} +(-90.9885 - 52.5322i) q^{71} +(-48.5118 - 84.0248i) q^{73} +(-33.7262 + 40.1933i) q^{77} +(-104.299 - 37.9616i) q^{79} +(44.8984 - 123.357i) q^{83} +(23.1965 + 19.4642i) q^{85} +(-87.8638 + 50.7282i) q^{89} +(6.99367 - 12.1134i) q^{91} +(42.5172 + 116.815i) q^{95} +(-28.7527 - 163.065i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 9 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 9 q^{5} - 36 q^{11} + 18 q^{23} - 9 q^{25} + 18 q^{29} + 45 q^{31} + 243 q^{35} + 198 q^{41} + 90 q^{43} + 243 q^{47} + 72 q^{49} - 252 q^{59} - 144 q^{61} - 747 q^{65} + 108 q^{67} - 324 q^{71} - 63 q^{73} - 495 q^{77} + 36 q^{79} + 27 q^{83} - 180 q^{85} + 567 q^{89} + 99 q^{91} + 1044 q^{95} - 216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{18}\right)\)

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
\(5\) 4.32861 + 0.763252i 0.865723 + 0.152650i 0.588836 0.808252i \(-0.299586\pi\)
0.276887 + 0.960903i \(0.410697\pi\)
\(6\) 0 0
\(7\) −2.73772 + 2.29722i −0.391103 + 0.328175i −0.817043 0.576577i \(-0.804388\pi\)
0.425939 + 0.904752i \(0.359944\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14.4582 2.54938i 1.31438 0.231762i 0.527866 0.849328i \(-0.322992\pi\)
0.786519 + 0.617566i \(0.211881\pi\)
\(12\) 0 0
\(13\) −3.67778 + 1.33860i −0.282906 + 0.102969i −0.479576 0.877500i \(-0.659209\pi\)
0.196670 + 0.980470i \(0.436987\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.96626 + 3.44462i 0.350956 + 0.202625i 0.665106 0.746749i \(-0.268386\pi\)
−0.314150 + 0.949373i \(0.601720\pi\)
\(18\) 0 0
\(19\) 14.1412 + 24.4932i 0.744272 + 1.28912i 0.950534 + 0.310620i \(0.100537\pi\)
−0.206263 + 0.978497i \(0.566130\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.832583 + 0.992234i −0.0361993 + 0.0431406i −0.783840 0.620962i \(-0.786742\pi\)
0.747641 + 0.664103i \(0.231186\pi\)
\(24\) 0 0
\(25\) −5.33796 1.94286i −0.213519 0.0777144i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 12.9959 35.7059i 0.448134 1.23124i −0.485887 0.874021i \(-0.661504\pi\)
0.934021 0.357217i \(-0.116274\pi\)
\(30\) 0 0
\(31\) 41.7175 + 35.0052i 1.34573 + 1.12920i 0.980115 + 0.198431i \(0.0635846\pi\)
0.365612 + 0.930767i \(0.380860\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −13.6039 + 7.85422i −0.388683 + 0.224406i
\(36\) 0 0
\(37\) 18.5334 32.1007i 0.500902 0.867588i −0.499097 0.866546i \(-0.666335\pi\)
0.999999 0.00104187i \(-0.000331638\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 14.0768 + 38.6757i 0.343336 + 0.943309i 0.984419 + 0.175837i \(0.0562632\pi\)
−0.641083 + 0.767472i \(0.721515\pi\)
\(42\) 0 0
\(43\) −0.615899 3.49294i −0.0143232 0.0812311i 0.976808 0.214116i \(-0.0686870\pi\)
−0.991131 + 0.132885i \(0.957576\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 27.5222 + 32.7997i 0.585578 + 0.697865i 0.974750 0.223300i \(-0.0716830\pi\)
−0.389171 + 0.921165i \(0.627239\pi\)
\(48\) 0 0
\(49\) −6.29086 + 35.6773i −0.128385 + 0.728107i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 47.8007i 0.901900i −0.892549 0.450950i \(-0.851085\pi\)
0.892549 0.450950i \(-0.148915\pi\)
\(54\) 0 0
\(55\) 64.5299 1.17327
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 61.1091 + 10.7752i 1.03575 + 0.182630i 0.665574 0.746332i \(-0.268187\pi\)
0.370174 + 0.928962i \(0.379298\pi\)
\(60\) 0 0
\(61\) −8.50896 + 7.13987i −0.139491 + 0.117047i −0.709863 0.704339i \(-0.751243\pi\)
0.570372 + 0.821386i \(0.306799\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −16.9414 + 2.98722i −0.260636 + 0.0459572i
\(66\) 0 0
\(67\) −105.153 + 38.2727i −1.56945 + 0.571235i −0.972877 0.231321i \(-0.925695\pi\)
−0.596577 + 0.802556i \(0.703473\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −90.9885 52.5322i −1.28153 0.739890i −0.304400 0.952544i \(-0.598456\pi\)
−0.977128 + 0.212654i \(0.931789\pi\)
\(72\) 0 0
\(73\) −48.5118 84.0248i −0.664545 1.15102i −0.979409 0.201888i \(-0.935292\pi\)
0.314864 0.949137i \(-0.398041\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −33.7262 + 40.1933i −0.438002 + 0.521991i
\(78\) 0 0
\(79\) −104.299 37.9616i −1.32024 0.480527i −0.416702 0.909043i \(-0.636814\pi\)
−0.903534 + 0.428516i \(0.859037\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 44.8984 123.357i 0.540945 1.48623i −0.304679 0.952455i \(-0.598549\pi\)
0.845624 0.533779i \(-0.179229\pi\)
\(84\) 0 0
\(85\) 23.1965 + 19.4642i 0.272900 + 0.228990i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −87.8638 + 50.7282i −0.987233 + 0.569979i −0.904446 0.426588i \(-0.859715\pi\)
−0.0827871 + 0.996567i \(0.526382\pi\)
\(90\) 0 0
\(91\) 6.99367 12.1134i 0.0768535 0.133114i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 42.5172 + 116.815i 0.447549 + 1.22963i
\(96\) 0 0
\(97\) −28.7527 163.065i −0.296420 1.68108i −0.661376 0.750055i \(-0.730027\pi\)
0.364956 0.931025i \(-0.381084\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.2076 16.9319i −0.140669 0.167643i 0.691110 0.722749i \(-0.257122\pi\)
−0.831779 + 0.555107i \(0.812677\pi\)
\(102\) 0 0
\(103\) −4.91300 + 27.8630i −0.0476991 + 0.270515i −0.999325 0.0367451i \(-0.988301\pi\)
0.951626 + 0.307260i \(0.0994121\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 67.0830i 0.626944i 0.949597 + 0.313472i \(0.101492\pi\)
−0.949597 + 0.313472i \(0.898508\pi\)
\(108\) 0 0
\(109\) 82.0479 0.752733 0.376367 0.926471i \(-0.377173\pi\)
0.376367 + 0.926471i \(0.377173\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −139.573 24.6104i −1.23516 0.217791i −0.482317 0.875997i \(-0.660205\pi\)
−0.752839 + 0.658205i \(0.771316\pi\)
\(114\) 0 0
\(115\) −4.36126 + 3.65953i −0.0379240 + 0.0318220i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −24.2470 + 4.27540i −0.203757 + 0.0359278i
\(120\) 0 0
\(121\) 88.8384 32.3345i 0.734202 0.267228i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −116.786 67.4265i −0.934289 0.539412i
\(126\) 0 0
\(127\) 36.1258 + 62.5718i 0.284455 + 0.492691i 0.972477 0.232999i \(-0.0748539\pi\)
−0.688022 + 0.725690i \(0.741521\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −80.4893 + 95.9234i −0.614422 + 0.732239i −0.980101 0.198502i \(-0.936392\pi\)
0.365679 + 0.930741i \(0.380837\pi\)
\(132\) 0 0
\(133\) −94.9809 34.5702i −0.714142 0.259927i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 44.4879 122.229i 0.324729 0.892186i −0.664693 0.747117i \(-0.731437\pi\)
0.989422 0.145069i \(-0.0463403\pi\)
\(138\) 0 0
\(139\) −64.7192 54.3058i −0.465606 0.390689i 0.379583 0.925158i \(-0.376067\pi\)
−0.845189 + 0.534468i \(0.820512\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −49.7615 + 28.7298i −0.347983 + 0.200908i
\(144\) 0 0
\(145\) 83.5068 144.638i 0.575909 0.997503i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −48.8305 134.161i −0.327722 0.900408i −0.988687 0.149992i \(-0.952075\pi\)
0.660966 0.750416i \(-0.270147\pi\)
\(150\) 0 0
\(151\) 7.41354 + 42.0443i 0.0490963 + 0.278439i 0.999466 0.0326849i \(-0.0104058\pi\)
−0.950369 + 0.311124i \(0.899295\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 153.861 + 183.365i 0.992654 + 1.18300i
\(156\) 0 0
\(157\) −11.8042 + 66.9447i −0.0751857 + 0.426399i 0.923860 + 0.382730i \(0.125016\pi\)
−0.999046 + 0.0436696i \(0.986095\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.62909i 0.0287521i
\(162\) 0 0
\(163\) −52.2555 −0.320586 −0.160293 0.987069i \(-0.551244\pi\)
−0.160293 + 0.987069i \(0.551244\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 205.761 + 36.2813i 1.23210 + 0.217253i 0.751529 0.659700i \(-0.229317\pi\)
0.480575 + 0.876953i \(0.340428\pi\)
\(168\) 0 0
\(169\) −117.727 + 98.7850i −0.696611 + 0.584526i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −285.514 + 50.3439i −1.65037 + 0.291005i −0.919963 0.392005i \(-0.871782\pi\)
−0.730409 + 0.683010i \(0.760671\pi\)
\(174\) 0 0
\(175\) 19.0771 6.94348i 0.109012 0.0396770i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 84.3915 + 48.7235i 0.471461 + 0.272198i 0.716851 0.697226i \(-0.245583\pi\)
−0.245390 + 0.969424i \(0.578916\pi\)
\(180\) 0 0
\(181\) 18.6602 + 32.3205i 0.103095 + 0.178566i 0.912958 0.408053i \(-0.133792\pi\)
−0.809863 + 0.586619i \(0.800459\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 104.725 124.806i 0.566080 0.674628i
\(186\) 0 0
\(187\) 95.0432 + 34.5929i 0.508252 + 0.184989i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 119.816 329.191i 0.627307 1.72351i −0.0610446 0.998135i \(-0.519443\pi\)
0.688352 0.725377i \(-0.258335\pi\)
\(192\) 0 0
\(193\) 8.46714 + 7.10477i 0.0438712 + 0.0368123i 0.664460 0.747324i \(-0.268662\pi\)
−0.620588 + 0.784136i \(0.713106\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 275.483 159.050i 1.39839 0.807362i 0.404167 0.914685i \(-0.367561\pi\)
0.994224 + 0.107323i \(0.0342280\pi\)
\(198\) 0 0
\(199\) −88.3486 + 153.024i −0.443963 + 0.768966i −0.997979 0.0635397i \(-0.979761\pi\)
0.554017 + 0.832506i \(0.313094\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 46.4453 + 127.607i 0.228794 + 0.628608i
\(204\) 0 0
\(205\) 31.4137 + 178.156i 0.153238 + 0.869054i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 266.899 + 318.077i 1.27703 + 1.52190i
\(210\) 0 0
\(211\) −36.0058 + 204.199i −0.170644 + 0.967768i 0.772409 + 0.635125i \(0.219051\pi\)
−0.943053 + 0.332643i \(0.892060\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 15.5897i 0.0725100i
\(216\) 0 0
\(217\) −194.626 −0.896893
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −26.5535 4.68210i −0.120152 0.0211860i
\(222\) 0 0
\(223\) 316.844 265.863i 1.42082 1.19221i 0.469926 0.882706i \(-0.344281\pi\)
0.950897 0.309506i \(-0.100164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −94.9127 + 16.7357i −0.418117 + 0.0737254i −0.378749 0.925499i \(-0.623646\pi\)
−0.0393683 + 0.999225i \(0.512535\pi\)
\(228\) 0 0
\(229\) 189.133 68.8386i 0.825906 0.300605i 0.105729 0.994395i \(-0.466283\pi\)
0.720178 + 0.693790i \(0.244060\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 303.285 + 175.101i 1.30165 + 0.751508i 0.980687 0.195583i \(-0.0626600\pi\)
0.320964 + 0.947092i \(0.395993\pi\)
\(234\) 0 0
\(235\) 94.0985 + 162.983i 0.400419 + 0.693547i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 100.663 119.965i 0.421183 0.501947i −0.513174 0.858285i \(-0.671530\pi\)
0.934357 + 0.356338i \(0.115975\pi\)
\(240\) 0 0
\(241\) 324.419 + 118.079i 1.34614 + 0.489954i 0.911740 0.410768i \(-0.134739\pi\)
0.434397 + 0.900721i \(0.356961\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −54.4614 + 149.632i −0.222292 + 0.610741i
\(246\) 0 0
\(247\) −84.7946 71.1512i −0.343298 0.288061i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −22.8819 + 13.2109i −0.0911628 + 0.0526329i −0.544888 0.838509i \(-0.683428\pi\)
0.453726 + 0.891141i \(0.350095\pi\)
\(252\) 0 0
\(253\) −9.50810 + 16.4685i −0.0375814 + 0.0650930i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −40.6520 111.691i −0.158179 0.434593i 0.835134 0.550047i \(-0.185390\pi\)
−0.993313 + 0.115453i \(0.963168\pi\)
\(258\) 0 0
\(259\) 23.0033 + 130.458i 0.0888159 + 0.503700i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −110.151 131.272i −0.418823 0.499134i 0.514840 0.857286i \(-0.327851\pi\)
−0.933663 + 0.358152i \(0.883407\pi\)
\(264\) 0 0
\(265\) 36.4840 206.911i 0.137675 0.780796i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 147.131i 0.546957i −0.961878 0.273478i \(-0.911826\pi\)
0.961878 0.273478i \(-0.0881742\pi\)
\(270\) 0 0
\(271\) 42.2256 0.155814 0.0779071 0.996961i \(-0.475176\pi\)
0.0779071 + 0.996961i \(0.475176\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −82.1306 14.4818i −0.298657 0.0526612i
\(276\) 0 0
\(277\) −241.990 + 203.054i −0.873611 + 0.733046i −0.964855 0.262782i \(-0.915360\pi\)
0.0912446 + 0.995829i \(0.470916\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −425.296 + 74.9912i −1.51351 + 0.266872i −0.867878 0.496777i \(-0.834517\pi\)
−0.645631 + 0.763650i \(0.723406\pi\)
\(282\) 0 0
\(283\) −51.4917 + 18.7414i −0.181949 + 0.0662242i −0.431388 0.902166i \(-0.641976\pi\)
0.249439 + 0.968391i \(0.419754\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −127.385 73.5457i −0.443850 0.256257i
\(288\) 0 0
\(289\) −120.769 209.178i −0.417886 0.723801i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 90.3507 107.676i 0.308364 0.367494i −0.589499 0.807769i \(-0.700675\pi\)
0.897863 + 0.440275i \(0.145119\pi\)
\(294\) 0 0
\(295\) 256.294 + 93.2833i 0.868792 + 0.316214i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.73385 4.76371i 0.00579883 0.0159321i
\(300\) 0 0
\(301\) 9.71021 + 8.14784i 0.0322598 + 0.0270692i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −42.2815 + 24.4113i −0.138628 + 0.0800369i
\(306\) 0 0
\(307\) −213.941 + 370.556i −0.696876 + 1.20702i 0.272669 + 0.962108i \(0.412094\pi\)
−0.969544 + 0.244916i \(0.921240\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −142.432 391.327i −0.457979 1.25829i −0.926987 0.375094i \(-0.877611\pi\)
0.469008 0.883194i \(-0.344612\pi\)
\(312\) 0 0
\(313\) 25.1171 + 142.446i 0.0802462 + 0.455099i 0.998282 + 0.0586003i \(0.0186638\pi\)
−0.918035 + 0.396499i \(0.870225\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 348.939 + 415.849i 1.10075 + 1.31183i 0.946106 + 0.323858i \(0.104980\pi\)
0.154648 + 0.987970i \(0.450576\pi\)
\(318\) 0 0
\(319\) 96.8698 549.376i 0.303667 1.72218i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 194.844i 0.603231i
\(324\) 0 0
\(325\) 22.2325 0.0684078
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −150.696 26.5718i −0.458043 0.0807654i
\(330\) 0 0
\(331\) 169.745 142.433i 0.512825 0.430311i −0.349297 0.937012i \(-0.613580\pi\)
0.862122 + 0.506701i \(0.169135\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −484.380 + 85.4093i −1.44591 + 0.254953i
\(336\) 0 0
\(337\) 36.2922 13.2093i 0.107692 0.0391967i −0.287612 0.957747i \(-0.592861\pi\)
0.395304 + 0.918550i \(0.370639\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 692.403 + 399.759i 2.03051 + 1.17231i
\(342\) 0 0
\(343\) −152.295 263.783i −0.444009 0.769046i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −134.499 + 160.290i −0.387607 + 0.461932i −0.924200 0.381910i \(-0.875267\pi\)
0.536593 + 0.843841i \(0.319711\pi\)
\(348\) 0 0
\(349\) −262.851 95.6701i −0.753156 0.274126i −0.0632228 0.997999i \(-0.520138\pi\)
−0.689933 + 0.723873i \(0.742360\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.7312 78.9384i 0.0813916 0.223622i −0.892321 0.451401i \(-0.850924\pi\)
0.973713 + 0.227780i \(0.0731466\pi\)
\(354\) 0 0
\(355\) −353.759 296.839i −0.996503 0.836166i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 221.262 127.746i 0.616330 0.355838i −0.159109 0.987261i \(-0.550862\pi\)
0.775439 + 0.631423i \(0.217529\pi\)
\(360\) 0 0
\(361\) −219.445 + 380.090i −0.607880 + 1.05288i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −145.857 400.738i −0.399607 1.09791i
\(366\) 0 0
\(367\) −44.7221 253.632i −0.121859 0.691095i −0.983124 0.182940i \(-0.941439\pi\)
0.861265 0.508155i \(-0.169672\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 109.809 + 130.865i 0.295981 + 0.352736i
\(372\) 0 0
\(373\) 83.5846 474.032i 0.224087 1.27086i −0.640335 0.768096i \(-0.721204\pi\)
0.864422 0.502767i \(-0.167684\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 148.715i 0.394469i
\(378\) 0 0
\(379\) −370.571 −0.977759 −0.488880 0.872351i \(-0.662594\pi\)
−0.488880 + 0.872351i \(0.662594\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −481.807 84.9556i −1.25798 0.221816i −0.495375 0.868679i \(-0.664969\pi\)
−0.762606 + 0.646863i \(0.776081\pi\)
\(384\) 0 0
\(385\) −176.665 + 148.240i −0.458870 + 0.385038i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −309.685 + 54.6057i −0.796104 + 0.140375i −0.556884 0.830591i \(-0.688003\pi\)
−0.239221 + 0.970965i \(0.576892\pi\)
\(390\) 0 0
\(391\) −8.38528 + 3.05199i −0.0214457 + 0.00780560i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −422.495 243.927i −1.06961 0.617538i
\(396\) 0 0
\(397\) −44.8022 77.5996i −0.112852 0.195465i 0.804067 0.594538i \(-0.202665\pi\)
−0.916919 + 0.399073i \(0.869332\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 55.2456 65.8392i 0.137770 0.164187i −0.692748 0.721180i \(-0.743600\pi\)
0.830518 + 0.556992i \(0.188045\pi\)
\(402\) 0 0
\(403\) −200.286 72.8980i −0.496987 0.180888i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 186.123 511.369i 0.457305 1.25643i
\(408\) 0 0
\(409\) 479.987 + 402.757i 1.17356 + 0.984736i 1.00000 8.63705e-5i \(-2.74926e-5\pi\)
0.173563 + 0.984823i \(0.444472\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −192.053 + 110.882i −0.465019 + 0.268479i
\(414\) 0 0
\(415\) 288.501 499.698i 0.695183 1.20409i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 120.491 + 331.047i 0.287569 + 0.790089i 0.996405 + 0.0847157i \(0.0269982\pi\)
−0.708836 + 0.705373i \(0.750780\pi\)
\(420\) 0 0
\(421\) 26.8471 + 152.258i 0.0637699 + 0.361657i 0.999949 + 0.0101329i \(0.00322545\pi\)
−0.936179 + 0.351524i \(0.885663\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −25.1553 29.9789i −0.0591888 0.0705385i
\(426\) 0 0
\(427\) 6.89332 39.0940i 0.0161436 0.0915549i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 226.253i 0.524949i 0.964939 + 0.262474i \(0.0845386\pi\)
−0.964939 + 0.262474i \(0.915461\pi\)
\(432\) 0 0
\(433\) −648.539 −1.49778 −0.748890 0.662695i \(-0.769413\pi\)
−0.748890 + 0.662695i \(0.769413\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −36.0767 6.36129i −0.0825553 0.0145567i
\(438\) 0 0
\(439\) 231.313 194.095i 0.526909 0.442129i −0.340123 0.940381i \(-0.610469\pi\)
0.867032 + 0.498252i \(0.166024\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −458.869 + 80.9110i −1.03582 + 0.182643i −0.665606 0.746303i \(-0.731827\pi\)
−0.370215 + 0.928946i \(0.620716\pi\)
\(444\) 0 0
\(445\) −419.047 + 152.521i −0.941678 + 0.342743i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −35.8903 20.7213i −0.0799339 0.0461498i 0.459500 0.888178i \(-0.348029\pi\)
−0.539434 + 0.842028i \(0.681362\pi\)
\(450\) 0 0
\(451\) 302.124 + 523.295i 0.669899 + 1.16030i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 39.5185 47.0963i 0.0868538 0.103508i
\(456\) 0 0
\(457\) 405.966 + 147.760i 0.888329 + 0.323325i 0.745566 0.666432i \(-0.232179\pi\)
0.142763 + 0.989757i \(0.454401\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −224.994 + 618.167i −0.488057 + 1.34093i 0.414380 + 0.910104i \(0.363998\pi\)
−0.902437 + 0.430822i \(0.858224\pi\)
\(462\) 0 0
\(463\) −415.550 348.688i −0.897516 0.753105i 0.0721872 0.997391i \(-0.477002\pi\)
−0.969703 + 0.244286i \(0.921447\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 71.4291 41.2396i 0.152953 0.0883075i −0.421570 0.906796i \(-0.638521\pi\)
0.574523 + 0.818488i \(0.305188\pi\)
\(468\) 0 0
\(469\) 199.960 346.341i 0.426354 0.738467i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −17.8096 48.9315i −0.0376525 0.103449i
\(474\) 0 0
\(475\) −27.8981 158.218i −0.0587329 0.333091i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −54.1626 64.5485i −0.113074 0.134757i 0.706538 0.707675i \(-0.250256\pi\)
−0.819613 + 0.572918i \(0.805811\pi\)
\(480\) 0 0
\(481\) −25.1915 + 142.868i −0.0523732 + 0.297023i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 727.790i 1.50060i
\(486\) 0 0
\(487\) 566.361 1.16296 0.581479 0.813561i \(-0.302474\pi\)
0.581479 + 0.813561i \(0.302474\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 754.890 + 133.107i 1.53745 + 0.271095i 0.877266 0.480005i \(-0.159365\pi\)
0.660188 + 0.751100i \(0.270476\pi\)
\(492\) 0 0
\(493\) 200.530 168.265i 0.406755 0.341308i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 369.779 65.2021i 0.744023 0.131191i
\(498\) 0 0
\(499\) −207.387 + 75.4828i −0.415606 + 0.151268i −0.541355 0.840794i \(-0.682089\pi\)
0.125750 + 0.992062i \(0.459866\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −783.399 452.296i −1.55745 0.899196i −0.997500 0.0706721i \(-0.977486\pi\)
−0.559954 0.828524i \(-0.689181\pi\)
\(504\) 0 0
\(505\) −48.5757 84.1357i −0.0961896 0.166605i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −220.212 + 262.439i −0.432637 + 0.515596i −0.937681 0.347497i \(-0.887032\pi\)
0.505044 + 0.863093i \(0.331476\pi\)
\(510\) 0 0
\(511\) 325.835 + 118.594i 0.637643 + 0.232083i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −42.5330 + 116.858i −0.0825884 + 0.226910i
\(516\) 0 0
\(517\) 481.541 + 404.061i 0.931414 + 0.781549i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 120.568 69.6099i 0.231416 0.133608i −0.379809 0.925065i \(-0.624010\pi\)
0.611225 + 0.791457i \(0.290677\pi\)
\(522\) 0 0
\(523\) −12.3821 + 21.4464i −0.0236751 + 0.0410065i −0.877620 0.479357i \(-0.840870\pi\)
0.853945 + 0.520363i \(0.174203\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 128.318 + 352.551i 0.243488 + 0.668977i
\(528\) 0 0
\(529\) 91.5686 + 519.311i 0.173097 + 0.981684i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −103.543 123.397i −0.194264 0.231514i
\(534\) 0 0
\(535\) −51.2012 + 290.376i −0.0957032 + 0.542760i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 531.868i 0.986768i
\(540\) 0 0
\(541\) 476.086 0.880011 0.440006 0.897995i \(-0.354976\pi\)
0.440006 + 0.897995i \(0.354976\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 355.154 + 62.6232i 0.651658 + 0.114905i
\(546\) 0 0
\(547\) −225.822 + 189.487i −0.412836 + 0.346411i −0.825430 0.564504i \(-0.809067\pi\)
0.412594 + 0.910915i \(0.364623\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1058.33 186.612i 1.92074 0.338679i
\(552\) 0 0
\(553\) 372.747 135.669i 0.674046 0.245333i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 756.664 + 436.860i 1.35846 + 0.784309i 0.989417 0.145101i \(-0.0463509\pi\)
0.369047 + 0.929411i \(0.379684\pi\)
\(558\) 0 0
\(559\) 6.94078 + 12.0218i 0.0124164 + 0.0215059i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 327.795 390.651i 0.582230 0.693874i −0.391863 0.920024i \(-0.628169\pi\)
0.974093 + 0.226149i \(0.0726137\pi\)
\(564\) 0 0
\(565\) −585.372 213.058i −1.03606 0.377094i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 303.435 833.680i 0.533277 1.46517i −0.321871 0.946784i \(-0.604312\pi\)
0.855148 0.518384i \(-0.173466\pi\)
\(570\) 0 0
\(571\) 403.162 + 338.293i 0.706063 + 0.592457i 0.923491 0.383619i \(-0.125323\pi\)
−0.217429 + 0.976076i \(0.569767\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.37207 3.67892i 0.0110819 0.00639812i
\(576\) 0 0
\(577\) 68.5779 118.780i 0.118853 0.205859i −0.800461 0.599385i \(-0.795412\pi\)
0.919313 + 0.393527i \(0.128745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 160.460 + 440.860i 0.276179 + 0.758796i
\(582\) 0 0
\(583\) −121.862 691.114i −0.209026 1.18544i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 115.043 + 137.103i 0.195985 + 0.233566i 0.855083 0.518491i \(-0.173506\pi\)
−0.659098 + 0.752057i \(0.729062\pi\)
\(588\) 0 0
\(589\) −267.454 + 1516.81i −0.454082 + 2.57523i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 52.0285i 0.0877377i −0.999037 0.0438689i \(-0.986032\pi\)
0.999037 0.0438689i \(-0.0139684\pi\)
\(594\) 0 0
\(595\) −108.219 −0.181881
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −114.494 20.1884i −0.191142 0.0337035i 0.0772575 0.997011i \(-0.475384\pi\)
−0.268400 + 0.963308i \(0.586495\pi\)
\(600\) 0 0
\(601\) −402.677 + 337.886i −0.670011 + 0.562206i −0.913069 0.407806i \(-0.866294\pi\)
0.243057 + 0.970012i \(0.421850\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 409.227 72.1577i 0.676408 0.119269i
\(606\) 0 0
\(607\) 80.4769 29.2912i 0.132581 0.0482557i −0.274877 0.961479i \(-0.588637\pi\)
0.407459 + 0.913224i \(0.366415\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −145.126 83.7886i −0.237522 0.137134i
\(612\) 0 0
\(613\) 81.6463 + 141.416i 0.133191 + 0.230694i 0.924905 0.380198i \(-0.124144\pi\)
−0.791714 + 0.610892i \(0.790811\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −179.054 + 213.388i −0.290200 + 0.345847i −0.891372 0.453273i \(-0.850256\pi\)
0.601171 + 0.799120i \(0.294701\pi\)
\(618\) 0 0
\(619\) −669.731 243.762i −1.08196 0.393800i −0.261321 0.965252i \(-0.584158\pi\)
−0.820636 + 0.571452i \(0.806380\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 124.013 340.722i 0.199057 0.546906i
\(624\) 0 0
\(625\) −345.270 289.716i −0.552432 0.463545i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 221.150 127.681i 0.351590 0.202990i
\(630\) 0 0
\(631\) −311.029 + 538.718i −0.492914 + 0.853753i −0.999967 0.00816241i \(-0.997402\pi\)
0.507052 + 0.861915i \(0.330735\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 108.617 + 298.422i 0.171050 + 0.469956i
\(636\) 0 0
\(637\) −24.6212 139.634i −0.0386518 0.219206i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −163.405 194.738i −0.254921 0.303804i 0.623372 0.781925i \(-0.285762\pi\)
−0.878294 + 0.478122i \(0.841318\pi\)
\(642\) 0 0
\(643\) 36.8378 208.917i 0.0572905 0.324910i −0.942671 0.333724i \(-0.891695\pi\)
0.999961 + 0.00881399i \(0.00280562\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 330.267i 0.510459i −0.966880 0.255230i \(-0.917849\pi\)
0.966880 0.255230i \(-0.0821511\pi\)
\(648\) 0 0
\(649\) 911.000 1.40370
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −329.228 58.0518i −0.504178 0.0889002i −0.0842262 0.996447i \(-0.526842\pi\)
−0.419952 + 0.907546i \(0.637953\pi\)
\(654\) 0 0
\(655\) −421.621 + 353.782i −0.643696 + 0.540125i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1021.20 180.064i 1.54962 0.273239i 0.667622 0.744501i \(-0.267312\pi\)
0.881993 + 0.471262i \(0.156201\pi\)
\(660\) 0 0
\(661\) 397.057 144.517i 0.600692 0.218634i −0.0237335 0.999718i \(-0.507555\pi\)
0.624426 + 0.781084i \(0.285333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −384.750 222.136i −0.578572 0.334038i
\(666\) 0 0
\(667\) 24.6085 + 42.6231i 0.0368942 + 0.0639027i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −104.822 + 124.922i −0.156218 + 0.186173i
\(672\) 0 0
\(673\) −14.3659 5.22876i −0.0213460 0.00776932i 0.331325 0.943517i \(-0.392504\pi\)
−0.352671 + 0.935747i \(0.614727\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −103.653 + 284.784i −0.153106 + 0.420655i −0.992405 0.123015i \(-0.960744\pi\)
0.839299 + 0.543670i \(0.182966\pi\)
\(678\) 0 0
\(679\) 453.313 + 380.375i 0.667618 + 0.560198i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −688.686 + 397.613i −1.00833 + 0.582157i −0.910701 0.413067i \(-0.864458\pi\)
−0.0976243 + 0.995223i \(0.531124\pi\)
\(684\) 0 0
\(685\) 285.863 495.129i 0.417318 0.722816i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 63.9861 + 175.800i 0.0928680 + 0.255153i
\(690\) 0 0
\(691\) 8.31097 + 47.1339i 0.0120275 + 0.0682111i 0.990231 0.139438i \(-0.0445297\pi\)
−0.978203 + 0.207650i \(0.933419\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −238.695 284.466i −0.343447 0.409304i
\(696\) 0 0
\(697\) −49.2372 + 279.238i −0.0706416 + 0.400629i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.9301i 0.0469759i 0.999724 + 0.0234880i \(0.00747714\pi\)
−0.999724 + 0.0234880i \(0.992523\pi\)
\(702\) 0 0
\(703\) 1048.33 1.49123
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 77.7927 + 13.7170i 0.110032 + 0.0194016i
\(708\) 0 0
\(709\) 705.725 592.173i 0.995380 0.835223i 0.00904260 0.999959i \(-0.497122\pi\)
0.986338 + 0.164736i \(0.0526772\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −69.4666 + 12.2488i −0.0974286 + 0.0171793i
\(714\) 0 0
\(715\) −237.327 + 86.3798i −0.331925 + 0.120811i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −406.335 234.597i −0.565139 0.326283i 0.190067 0.981771i \(-0.439130\pi\)
−0.755205 + 0.655488i \(0.772463\pi\)
\(720\) 0 0
\(721\) −50.5571 87.5675i −0.0701209 0.121453i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −138.743 + 165.348i −0.191370 + 0.228066i
\(726\) 0 0
\(727\) 53.3053 + 19.4016i 0.0733223 + 0.0266871i 0.378421 0.925634i \(-0.376467\pi\)
−0.305099 + 0.952321i \(0.598689\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.35723 22.9613i 0.0114326 0.0314108i
\(732\) 0 0
\(733\) −773.970 649.438i −1.05589 0.886000i −0.0621924 0.998064i \(-0.519809\pi\)
−0.993701 + 0.112064i \(0.964254\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1422.76 + 821.432i −1.93048 + 1.11456i
\(738\) 0 0
\(739\) 368.682 638.576i 0.498893 0.864108i −0.501106 0.865386i \(-0.667073\pi\)
0.999999 + 0.00127777i \(0.000406727\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.4911 + 86.5211i 0.0423837 + 0.116448i 0.959079 0.283139i \(-0.0913757\pi\)
−0.916695 + 0.399587i \(0.869153\pi\)
\(744\) 0 0
\(745\) −108.970 618.000i −0.146269 0.829531i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −154.105 183.655i −0.205747 0.245200i
\(750\) 0 0
\(751\) −65.4270 + 371.055i −0.0871199 + 0.494081i 0.909759 + 0.415137i \(0.136266\pi\)
−0.996879 + 0.0789448i \(0.974845\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 187.652i 0.248545i
\(756\) 0 0
\(757\) 98.9786 0.130751 0.0653755 0.997861i \(-0.479175\pi\)
0.0653755 + 0.997861i \(0.479175\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1035.32 + 182.555i 1.36047 + 0.239888i 0.805804 0.592183i \(-0.201734\pi\)
0.554669 + 0.832071i \(0.312845\pi\)
\(762\) 0 0
\(763\) −224.625 + 188.482i −0.294396 + 0.247028i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −239.169 + 42.1720i −0.311824 + 0.0549830i
\(768\) 0 0
\(769\) 14.0610 5.11778i 0.0182848 0.00665511i −0.332862 0.942976i \(-0.608014\pi\)
0.351146 + 0.936321i \(0.385792\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −453.509 261.834i −0.586687 0.338724i 0.177099 0.984193i \(-0.443329\pi\)
−0.763787 + 0.645469i \(0.776662\pi\)
\(774\) 0 0
\(775\) −154.677 267.908i −0.199583 0.345687i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −748.229 + 891.705i −0.960499 + 1.14468i
\(780\) 0 0
\(781\) −1449.46 527.559i −1.85590 0.675492i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −102.191 + 280.768i −0.130180 + 0.357667i
\(786\) 0 0
\(787\) 120.730 + 101.305i 0.153406 + 0.128723i 0.716260 0.697833i \(-0.245852\pi\)
−0.562854 + 0.826556i \(0.690297\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 438.647 253.253i 0.554547 0.320168i
\(792\) 0 0
\(793\) 21.7366 37.6489i 0.0274106 0.0474766i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.59137 15.3622i −0.00701552 0.0192750i 0.936135 0.351640i \(-0.114376\pi\)
−0.943151 + 0.332365i \(0.892154\pi\)
\(798\) 0 0
\(799\) 51.2221 + 290.495i 0.0641077 + 0.363573i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −915.605 1091.18i −1.14023 1.35887i
\(804\) 0 0
\(805\) 3.53316 20.0376i 0.00438902 0.0248914i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1248.37i 1.54310i −0.636168 0.771551i \(-0.719481\pi\)
0.636168 0.771551i \(-0.280519\pi\)
\(810\) 0 0
\(811\) −796.242 −0.981803 −0.490901 0.871215i \(-0.663332\pi\)
−0.490901 + 0.871215i \(0.663332\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −226.194 39.8841i −0.277538 0.0489375i
\(816\) 0 0
\(817\) 76.8437 64.4795i 0.0940559 0.0789223i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1369.41 241.464i 1.66798 0.294110i 0.741640 0.670798i \(-0.234048\pi\)
0.926340 + 0.376688i \(0.122937\pi\)
\(822\) 0 0
\(823\) −21.0770 + 7.67138i −0.0256099 + 0.00932124i −0.354793 0.934945i \(-0.615449\pi\)
0.329183 + 0.944266i \(0.393227\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −79.4904 45.8938i −0.0961190 0.0554943i 0.451170 0.892438i \(-0.351007\pi\)
−0.547289 + 0.836944i \(0.684340\pi\)
\(828\) 0 0
\(829\) −398.805 690.751i −0.481068 0.833234i 0.518696 0.854959i \(-0.326418\pi\)
−0.999764 + 0.0217246i \(0.993084\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −160.428 + 191.190i −0.192590 + 0.229520i
\(834\) 0 0
\(835\) 862.970 + 314.095i 1.03350 + 0.376162i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −566.496 + 1556.43i −0.675203 + 1.85511i −0.187045 + 0.982351i \(0.559891\pi\)
−0.488158 + 0.872755i \(0.662331\pi\)
\(840\) 0 0
\(841\) −461.775 387.476i −0.549079 0.460732i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −584.994 + 337.746i −0.692301 + 0.399700i
\(846\) 0 0
\(847\) −168.935 + 292.605i −0.199451 + 0.345460i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 16.4209 + 45.1160i 0.0192960 + 0.0530153i
\(852\) 0 0
\(853\) −215.597 1222.71i −0.252752 1.43343i −0.801779 0.597621i \(-0.796113\pi\)
0.549027 0.835805i \(-0.314998\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −949.464 1131.53i −1.10789 1.32033i −0.942537 0.334101i \(-0.891567\pi\)
−0.165355 0.986234i \(-0.552877\pi\)
\(858\) 0 0
\(859\) −173.038 + 981.345i −0.201441 + 1.14243i 0.701503 + 0.712667i \(0.252513\pi\)
−0.902943 + 0.429760i \(0.858598\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 671.209i 0.777763i 0.921288 + 0.388881i \(0.127138\pi\)
−0.921288 + 0.388881i \(0.872862\pi\)
\(864\) 0 0
\(865\) −1274.31 −1.47319
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1604.75 282.961i −1.84667 0.325617i
\(870\) 0 0
\(871\) 335.499 281.517i 0.385188 0.323211i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 474.622 83.6886i 0.542425 0.0956441i
\(876\) 0 0
\(877\) −610.614 + 222.245i −0.696253 + 0.253415i −0.665810 0.746121i \(-0.731914\pi\)
−0.0304427 + 0.999537i \(0.509692\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 196.446 + 113.418i 0.222980 + 0.128738i 0.607330 0.794450i \(-0.292241\pi\)
−0.384349 + 0.923188i \(0.625574\pi\)
\(882\) 0 0
\(883\) 34.4995 + 59.7548i 0.0390707 + 0.0676725i 0.884900 0.465782i \(-0.154227\pi\)
−0.845829 + 0.533454i \(0.820894\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 581.417 692.906i 0.655488 0.781180i −0.331243 0.943545i \(-0.607468\pi\)
0.986731 + 0.162366i \(0.0519125\pi\)
\(888\) 0 0
\(889\) −242.644 88.3151i −0.272940 0.0993421i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −414.173 + 1137.93i −0.463800 + 1.27428i
\(894\) 0 0
\(895\) 328.110 + 275.317i 0.366603 + 0.307617i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1792.05 1034.64i 1.99338 1.15088i
\(900\) 0 0
\(901\) 164.655 285.191i 0.182747 0.316528i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 56.1043 + 154.145i 0.0619937 + 0.170326i
\(906\) 0 0
\(907\) −28.0243 158.934i −0.0308978 0.175230i 0.965454 0.260575i \(-0.0839120\pi\)
−0.996352 + 0.0853445i \(0.972801\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −51.9472 61.9082i −0.0570221 0.0679563i 0.736779 0.676133i \(-0.236346\pi\)
−0.793802 + 0.608177i \(0.791901\pi\)
\(912\) 0 0
\(913\) 334.668 1897.99i 0.366558 2.07885i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 447.513i 0.488019i
\(918\) 0 0
\(919\) −656.985 −0.714891 −0.357445 0.933934i \(-0.616352\pi\)
−0.357445 + 0.933934i \(0.616352\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 404.955 + 71.4045i 0.438738 + 0.0773613i
\(924\) 0 0
\(925\) −161.298 + 135.345i −0.174376 + 0.146319i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 244.546 43.1200i 0.263235 0.0464155i −0.0404727 0.999181i \(-0.512886\pi\)
0.303708 + 0.952765i \(0.401775\pi\)
\(930\) 0 0
\(931\) −962.811 + 350.434i −1.03417 + 0.376406i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 385.002 + 222.281i 0.411767 + 0.237734i
\(936\) 0 0
\(937\) 468.473 + 811.418i 0.499971 + 0.865975i 1.00000 3.38875e-5i \(-1.07867e-5\pi\)
−0.500029 + 0.866008i \(0.666677\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 899.429 1071.90i 0.955823 1.13911i −0.0343714 0.999409i \(-0.510943\pi\)
0.990194 0.139696i \(-0.0446127\pi\)
\(942\) 0 0
\(943\) −50.0954 18.2332i −0.0531234 0.0193354i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −249.635 + 685.866i −0.263606 + 0.724252i 0.735311 + 0.677730i \(0.237036\pi\)
−0.998917 + 0.0465220i \(0.985186\pi\)
\(948\) 0 0
\(949\) 290.891 + 244.087i 0.306524 + 0.257204i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1167.46 + 674.032i −1.22503 + 0.707274i −0.965987 0.258591i \(-0.916742\pi\)
−0.259047 + 0.965865i \(0.583409\pi\)
\(954\) 0 0
\(955\) 769.891 1333.49i 0.806169 1.39633i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 158.993 + 436.829i 0.165790 + 0.455505i
\(960\) 0 0
\(961\) 348.115 + 1974.26i 0.362242 + 2.05438i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 31.2283 + 37.2164i 0.0323609 + 0.0385662i
\(966\) 0 0
\(967\) 124.207 704.414i 0.128446 0.728453i −0.850755 0.525562i \(-0.823855\pi\)
0.979201 0.202891i \(-0.0650337\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1288.41i 1.32689i 0.748226 + 0.663444i \(0.230906\pi\)
−0.748226 + 0.663444i \(0.769094\pi\)
\(972\) 0 0
\(973\) 301.936 0.310314
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 830.119 + 146.372i 0.849661 + 0.149818i 0.581491 0.813553i \(-0.302470\pi\)
0.268171 + 0.963371i \(0.413581\pi\)
\(978\) 0 0
\(979\) −1141.03 + 957.437i −1.16551 + 0.977975i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −502.601 + 88.6222i −0.511293 + 0.0901548i −0.423342 0.905970i \(-0.639143\pi\)
−0.0879514 + 0.996125i \(0.528032\pi\)
\(984\) 0 0
\(985\) 1313.86 478.204i 1.33386 0.485487i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.97860 + 2.29704i 0.00402285 + 0.00232259i
\(990\) 0 0
\(991\) −186.798 323.544i −0.188495 0.326483i 0.756254 0.654278i \(-0.227028\pi\)
−0.944749 + 0.327796i \(0.893694\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −499.223 + 594.951i −0.501732 + 0.597940i
\(996\) 0 0
\(997\) 1763.31 + 641.793i 1.76862 + 0.643724i 0.999992 + 0.00402483i \(0.00128115\pi\)
0.768625 + 0.639699i \(0.220941\pi\)
\(998\) 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 324.3.k.a.17.4 36
3.2 odd 2 108.3.k.a.77.2 36
12.11 even 2 432.3.bc.b.401.5 36
27.7 even 9 108.3.k.a.101.2 yes 36
27.13 even 9 2916.3.c.b.1457.28 36
27.14 odd 18 2916.3.c.b.1457.9 36
27.20 odd 18 inner 324.3.k.a.305.4 36
108.7 odd 18 432.3.bc.b.209.5 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.77.2 36 3.2 odd 2
108.3.k.a.101.2 yes 36 27.7 even 9
324.3.k.a.17.4 36 1.1 even 1 trivial
324.3.k.a.305.4 36 27.20 odd 18 inner
432.3.bc.b.209.5 36 108.7 odd 18
432.3.bc.b.401.5 36 12.11 even 2
2916.3.c.b.1457.9 36 27.14 odd 18
2916.3.c.b.1457.28 36 27.13 even 9