Properties

Label 468.2.bz.a.401.1
Level $468$
Weight $2$
Character 468.401
Analytic conductor $3.737$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [468,2,Mod(41,468)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("468.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(468, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.bz (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 401.1
Character \(\chi\) \(=\) 468.401
Dual form 468.2.bz.a.461.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.71575 + 0.237081i) q^{3} +(-0.786297 - 2.93450i) q^{5} +(-0.474592 + 0.474592i) q^{7} +(2.88759 - 0.813543i) q^{9} +(-4.52204 + 1.21168i) q^{11} +(3.60335 + 0.125867i) q^{13} +(2.04480 + 4.84845i) q^{15} +(-3.61521 + 6.26173i) q^{17} +(-4.17267 + 1.11806i) q^{19} +(0.701763 - 0.926797i) q^{21} -1.46204 q^{23} +(-3.66289 + 2.11477i) q^{25} +(-4.76149 + 2.08043i) q^{27} +(-3.82932 - 2.21086i) q^{29} +(-1.38143 + 0.370153i) q^{31} +(7.47142 - 3.15102i) q^{33} +(1.76586 + 1.01952i) q^{35} +(-1.14370 - 0.306454i) q^{37} +(-6.21229 + 0.638331i) q^{39} +(-5.55355 + 5.55355i) q^{41} +9.46907i q^{43} +(-4.65784 - 7.83393i) q^{45} +(1.93010 - 7.20323i) q^{47} +6.54953i q^{49} +(4.71825 - 11.6006i) q^{51} -7.46793i q^{53} +(7.11133 + 12.3172i) q^{55} +(6.89419 - 2.90758i) q^{57} +(1.07072 - 3.99596i) q^{59} -5.61574 q^{61} +(-0.984323 + 1.75652i) q^{63} +(-2.46395 - 10.6730i) q^{65} +(-4.82399 - 4.82399i) q^{67} +(2.50850 - 0.346623i) q^{69} +(2.32349 + 8.67138i) q^{71} +(8.37710 - 8.37710i) q^{73} +(5.78323 - 4.49682i) q^{75} +(1.57107 - 2.72118i) q^{77} +(6.30240 + 10.9161i) q^{79} +(7.67630 - 4.69835i) q^{81} +(-4.29206 - 1.15005i) q^{83} +(21.2177 + 5.68525i) q^{85} +(7.09431 + 2.88542i) q^{87} +(-2.16797 + 8.09097i) q^{89} +(-1.76986 + 1.65039i) q^{91} +(2.28243 - 0.962601i) q^{93} +(6.56192 + 11.3656i) q^{95} +(-11.0288 - 11.0288i) q^{97} +(-12.0720 + 7.17770i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{7} + 12 q^{11} + 12 q^{15} + 4 q^{19} - 12 q^{21} - 4 q^{31} + 18 q^{33} + 66 q^{35} + 2 q^{37} + 24 q^{39} + 24 q^{41} + 24 q^{45} - 36 q^{47} - 12 q^{57} + 6 q^{63} - 36 q^{65} - 14 q^{67}+ \cdots - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) −1.71575 + 0.237081i −0.990588 + 0.136879i
\(4\) 0 0
\(5\) −0.786297 2.93450i −0.351642 1.31235i −0.884657 0.466242i \(-0.845608\pi\)
0.533015 0.846106i \(-0.321059\pi\)
\(6\) 0 0
\(7\) −0.474592 + 0.474592i −0.179379 + 0.179379i −0.791085 0.611706i \(-0.790483\pi\)
0.611706 + 0.791085i \(0.290483\pi\)
\(8\) 0 0
\(9\) 2.88759 0.813543i 0.962528 0.271181i
\(10\) 0 0
\(11\) −4.52204 + 1.21168i −1.36345 + 0.365335i −0.865081 0.501632i \(-0.832733\pi\)
−0.498366 + 0.866967i \(0.666066\pi\)
\(12\) 0 0
\(13\) 3.60335 + 0.125867i 0.999390 + 0.0349091i
\(14\) 0 0
\(15\) 2.04480 + 4.84845i 0.527965 + 1.25186i
\(16\) 0 0
\(17\) −3.61521 + 6.26173i −0.876817 + 1.51869i −0.0220024 + 0.999758i \(0.507004\pi\)
−0.854815 + 0.518934i \(0.826329\pi\)
\(18\) 0 0
\(19\) −4.17267 + 1.11806i −0.957277 + 0.256502i −0.703447 0.710747i \(-0.748357\pi\)
−0.253830 + 0.967249i \(0.581690\pi\)
\(20\) 0 0
\(21\) 0.701763 0.926797i 0.153137 0.202244i
\(22\) 0 0
\(23\) −1.46204 −0.304857 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(24\) 0 0
\(25\) −3.66289 + 2.11477i −0.732579 + 0.422954i
\(26\) 0 0
\(27\) −4.76149 + 2.08043i −0.916350 + 0.400378i
\(28\) 0 0
\(29\) −3.82932 2.21086i −0.711087 0.410546i 0.100376 0.994950i \(-0.467995\pi\)
−0.811463 + 0.584403i \(0.801329\pi\)
\(30\) 0 0
\(31\) −1.38143 + 0.370153i −0.248112 + 0.0664815i −0.380732 0.924686i \(-0.624328\pi\)
0.132619 + 0.991167i \(0.457661\pi\)
\(32\) 0 0
\(33\) 7.47142 3.15102i 1.30061 0.548523i
\(34\) 0 0
\(35\) 1.76586 + 1.01952i 0.298485 + 0.172330i
\(36\) 0 0
\(37\) −1.14370 0.306454i −0.188023 0.0503807i 0.163579 0.986530i \(-0.447696\pi\)
−0.351602 + 0.936150i \(0.614363\pi\)
\(38\) 0 0
\(39\) −6.21229 + 0.638331i −0.994762 + 0.102215i
\(40\) 0 0
\(41\) −5.55355 + 5.55355i −0.867319 + 0.867319i −0.992175 0.124856i \(-0.960153\pi\)
0.124856 + 0.992175i \(0.460153\pi\)
\(42\) 0 0
\(43\) 9.46907i 1.44402i 0.691883 + 0.722010i \(0.256782\pi\)
−0.691883 + 0.722010i \(0.743218\pi\)
\(44\) 0 0
\(45\) −4.65784 7.83393i −0.694350 1.16781i
\(46\) 0 0
\(47\) 1.93010 7.20323i 0.281534 1.05070i −0.669801 0.742541i \(-0.733621\pi\)
0.951335 0.308159i \(-0.0997128\pi\)
\(48\) 0 0
\(49\) 6.54953i 0.935646i
\(50\) 0 0
\(51\) 4.71825 11.6006i 0.660687 1.62441i
\(52\) 0 0
\(53\) 7.46793i 1.02580i −0.858449 0.512900i \(-0.828571\pi\)
0.858449 0.512900i \(-0.171429\pi\)
\(54\) 0 0
\(55\) 7.11133 + 12.3172i 0.958892 + 1.66085i
\(56\) 0 0
\(57\) 6.89419 2.90758i 0.913157 0.385118i
\(58\) 0 0
\(59\) 1.07072 3.99596i 0.139395 0.520230i −0.860546 0.509373i \(-0.829877\pi\)
0.999941 0.0108571i \(-0.00345598\pi\)
\(60\) 0 0
\(61\) −5.61574 −0.719022 −0.359511 0.933141i \(-0.617056\pi\)
−0.359511 + 0.933141i \(0.617056\pi\)
\(62\) 0 0
\(63\) −0.984323 + 1.75652i −0.124013 + 0.221301i
\(64\) 0 0
\(65\) −2.46395 10.6730i −0.305615 1.32382i
\(66\) 0 0
\(67\) −4.82399 4.82399i −0.589345 0.589345i 0.348109 0.937454i \(-0.386824\pi\)
−0.937454 + 0.348109i \(0.886824\pi\)
\(68\) 0 0
\(69\) 2.50850 0.346623i 0.301987 0.0417284i
\(70\) 0 0
\(71\) 2.32349 + 8.67138i 0.275747 + 1.02910i 0.955339 + 0.295513i \(0.0954904\pi\)
−0.679591 + 0.733591i \(0.737843\pi\)
\(72\) 0 0
\(73\) 8.37710 8.37710i 0.980466 0.980466i −0.0193471 0.999813i \(-0.506159\pi\)
0.999813 + 0.0193471i \(0.00615876\pi\)
\(74\) 0 0
\(75\) 5.78323 4.49682i 0.667790 0.519248i
\(76\) 0 0
\(77\) 1.57107 2.72118i 0.179040 0.310107i
\(78\) 0 0
\(79\) 6.30240 + 10.9161i 0.709076 + 1.22816i 0.965200 + 0.261512i \(0.0842210\pi\)
−0.256125 + 0.966644i \(0.582446\pi\)
\(80\) 0 0
\(81\) 7.67630 4.69835i 0.852922 0.522039i
\(82\) 0 0
\(83\) −4.29206 1.15005i −0.471114 0.126235i 0.0154478 0.999881i \(-0.495083\pi\)
−0.486562 + 0.873646i \(0.661749\pi\)
\(84\) 0 0
\(85\) 21.2177 + 5.68525i 2.30138 + 0.616652i
\(86\) 0 0
\(87\) 7.09431 + 2.88542i 0.760589 + 0.309349i
\(88\) 0 0
\(89\) −2.16797 + 8.09097i −0.229804 + 0.857641i 0.750618 + 0.660736i \(0.229756\pi\)
−0.980423 + 0.196905i \(0.936911\pi\)
\(90\) 0 0
\(91\) −1.76986 + 1.65039i −0.185531 + 0.173008i
\(92\) 0 0
\(93\) 2.28243 0.962601i 0.236677 0.0998170i
\(94\) 0 0
\(95\) 6.56192 + 11.3656i 0.673239 + 1.16608i
\(96\) 0 0
\(97\) −11.0288 11.0288i −1.11980 1.11980i −0.991770 0.128034i \(-0.959133\pi\)
−0.128034 0.991770i \(-0.540867\pi\)
\(98\) 0 0
\(99\) −12.0720 + 7.17770i −1.21328 + 0.721386i
\(100\) 0 0
\(101\) 4.88792 8.46612i 0.486366 0.842411i −0.513511 0.858083i \(-0.671655\pi\)
0.999877 + 0.0156721i \(0.00498879\pi\)
\(102\) 0 0
\(103\) −9.73205 5.61880i −0.958928 0.553637i −0.0630850 0.998008i \(-0.520094\pi\)
−0.895843 + 0.444371i \(0.853427\pi\)
\(104\) 0 0
\(105\) −3.27148 1.33059i −0.319264 0.129852i
\(106\) 0 0
\(107\) 14.6940 8.48360i 1.42052 0.820140i 0.424181 0.905577i \(-0.360562\pi\)
0.996344 + 0.0854370i \(0.0272286\pi\)
\(108\) 0 0
\(109\) 0.514307 + 0.514307i 0.0492617 + 0.0492617i 0.731309 0.682047i \(-0.238910\pi\)
−0.682047 + 0.731309i \(0.738910\pi\)
\(110\) 0 0
\(111\) 2.03496 + 0.254648i 0.193150 + 0.0241701i
\(112\) 0 0
\(113\) −8.52098 + 4.91959i −0.801586 + 0.462796i −0.844026 0.536303i \(-0.819820\pi\)
0.0424393 + 0.999099i \(0.486487\pi\)
\(114\) 0 0
\(115\) 1.14960 + 4.29036i 0.107201 + 0.400078i
\(116\) 0 0
\(117\) 10.5074 2.56803i 0.971408 0.237415i
\(118\) 0 0
\(119\) −1.25601 4.68751i −0.115139 0.429703i
\(120\) 0 0
\(121\) 9.45443 5.45852i 0.859493 0.496229i
\(122\) 0 0
\(123\) 8.21186 10.8451i 0.740438 0.977874i
\(124\) 0 0
\(125\) −1.65509 1.65509i −0.148036 0.148036i
\(126\) 0 0
\(127\) 12.2584 7.07742i 1.08776 0.628019i 0.154782 0.987949i \(-0.450533\pi\)
0.932980 + 0.359929i \(0.117199\pi\)
\(128\) 0 0
\(129\) −2.24494 16.2465i −0.197656 1.43043i
\(130\) 0 0
\(131\) −2.29027 1.32229i −0.200101 0.115529i 0.396601 0.917991i \(-0.370190\pi\)
−0.596703 + 0.802462i \(0.703523\pi\)
\(132\) 0 0
\(133\) 1.44969 2.51094i 0.125704 0.217726i
\(134\) 0 0
\(135\) 9.84896 + 12.3368i 0.847663 + 1.06178i
\(136\) 0 0
\(137\) −0.0685980 0.0685980i −0.00586072 0.00586072i 0.704170 0.710031i \(-0.251319\pi\)
−0.710031 + 0.704170i \(0.751319\pi\)
\(138\) 0 0
\(139\) −6.04452 10.4694i −0.512690 0.888004i −0.999892 0.0147152i \(-0.995316\pi\)
0.487202 0.873289i \(-0.338017\pi\)
\(140\) 0 0
\(141\) −1.60382 + 12.8165i −0.135066 + 1.07935i
\(142\) 0 0
\(143\) −16.4470 + 3.79693i −1.37537 + 0.317515i
\(144\) 0 0
\(145\) −3.47678 + 12.9755i −0.288731 + 1.07756i
\(146\) 0 0
\(147\) −1.55277 11.2373i −0.128070 0.926840i
\(148\) 0 0
\(149\) −12.1989 3.26870i −0.999377 0.267782i −0.278192 0.960525i \(-0.589735\pi\)
−0.721184 + 0.692743i \(0.756402\pi\)
\(150\) 0 0
\(151\) −14.4834 3.88081i −1.17864 0.315816i −0.384253 0.923228i \(-0.625541\pi\)
−0.794387 + 0.607412i \(0.792208\pi\)
\(152\) 0 0
\(153\) −5.34504 + 21.0224i −0.432121 + 1.69956i
\(154\) 0 0
\(155\) 2.17243 + 3.76276i 0.174494 + 0.302232i
\(156\) 0 0
\(157\) −1.33213 + 2.30731i −0.106315 + 0.184144i −0.914275 0.405094i \(-0.867239\pi\)
0.807960 + 0.589238i \(0.200572\pi\)
\(158\) 0 0
\(159\) 1.77051 + 12.8131i 0.140410 + 1.01614i
\(160\) 0 0
\(161\) 0.693873 0.693873i 0.0546849 0.0546849i
\(162\) 0 0
\(163\) −1.21282 4.52631i −0.0949955 0.354528i 0.902023 0.431687i \(-0.142082\pi\)
−0.997019 + 0.0771594i \(0.975415\pi\)
\(164\) 0 0
\(165\) −15.1214 19.4472i −1.17720 1.51397i
\(166\) 0 0
\(167\) −11.0290 11.0290i −0.853451 0.853451i 0.137106 0.990556i \(-0.456220\pi\)
−0.990556 + 0.137106i \(0.956220\pi\)
\(168\) 0 0
\(169\) 12.9683 + 0.907085i 0.997563 + 0.0697757i
\(170\) 0 0
\(171\) −11.1394 + 6.62316i −0.851848 + 0.506485i
\(172\) 0 0
\(173\) −20.0109 −1.52140 −0.760702 0.649101i \(-0.775145\pi\)
−0.760702 + 0.649101i \(0.775145\pi\)
\(174\) 0 0
\(175\) 0.734725 2.74203i 0.0555400 0.207278i
\(176\) 0 0
\(177\) −0.889710 + 7.10991i −0.0668747 + 0.534414i
\(178\) 0 0
\(179\) 5.74519 + 9.95097i 0.429416 + 0.743770i 0.996821 0.0796682i \(-0.0253861\pi\)
−0.567405 + 0.823439i \(0.692053\pi\)
\(180\) 0 0
\(181\) 26.1318i 1.94236i 0.238350 + 0.971179i \(0.423394\pi\)
−0.238350 + 0.971179i \(0.576606\pi\)
\(182\) 0 0
\(183\) 9.63519 1.33139i 0.712254 0.0984188i
\(184\) 0 0
\(185\) 3.59715i 0.264468i
\(186\) 0 0
\(187\) 8.76094 32.6963i 0.640663 2.39099i
\(188\) 0 0
\(189\) 1.27241 3.24712i 0.0925544 0.236193i
\(190\) 0 0
\(191\) 2.37903i 0.172141i 0.996289 + 0.0860703i \(0.0274310\pi\)
−0.996289 + 0.0860703i \(0.972569\pi\)
\(192\) 0 0
\(193\) −11.7424 + 11.7424i −0.845238 + 0.845238i −0.989535 0.144296i \(-0.953908\pi\)
0.144296 + 0.989535i \(0.453908\pi\)
\(194\) 0 0
\(195\) 6.75788 + 17.7280i 0.483942 + 1.26953i
\(196\) 0 0
\(197\) 23.6064 + 6.32532i 1.68189 + 0.450660i 0.968276 0.249882i \(-0.0803917\pi\)
0.713611 + 0.700542i \(0.247058\pi\)
\(198\) 0 0
\(199\) −11.8808 6.85940i −0.842210 0.486250i 0.0158050 0.999875i \(-0.494969\pi\)
−0.858015 + 0.513625i \(0.828302\pi\)
\(200\) 0 0
\(201\) 9.42044 + 7.13308i 0.664466 + 0.503129i
\(202\) 0 0
\(203\) 2.86662 0.768109i 0.201197 0.0539107i
\(204\) 0 0
\(205\) 20.6636 + 11.9302i 1.44321 + 0.833238i
\(206\) 0 0
\(207\) −4.22177 + 1.18943i −0.293433 + 0.0826714i
\(208\) 0 0
\(209\) 17.5143 10.1119i 1.21149 0.699453i
\(210\) 0 0
\(211\) −14.7322 −1.01420 −0.507102 0.861886i \(-0.669283\pi\)
−0.507102 + 0.861886i \(0.669283\pi\)
\(212\) 0 0
\(213\) −6.04234 14.3270i −0.414014 0.981673i
\(214\) 0 0
\(215\) 27.7870 7.44550i 1.89506 0.507779i
\(216\) 0 0
\(217\) 0.479944 0.831287i 0.0325807 0.0564314i
\(218\) 0 0
\(219\) −12.3869 + 16.3591i −0.837032 + 1.10544i
\(220\) 0 0
\(221\) −13.8150 + 22.1082i −0.929299 + 1.48716i
\(222\) 0 0
\(223\) 25.5686 6.85108i 1.71220 0.458782i 0.736236 0.676725i \(-0.236601\pi\)
0.975962 + 0.217942i \(0.0699345\pi\)
\(224\) 0 0
\(225\) −8.85646 + 9.08651i −0.590430 + 0.605767i
\(226\) 0 0
\(227\) 2.57813 2.57813i 0.171117 0.171117i −0.616353 0.787470i \(-0.711391\pi\)
0.787470 + 0.616353i \(0.211391\pi\)
\(228\) 0 0
\(229\) −3.18957 11.9036i −0.210773 0.786614i −0.987612 0.156915i \(-0.949845\pi\)
0.776840 0.629699i \(-0.216822\pi\)
\(230\) 0 0
\(231\) −2.05042 + 5.04132i −0.134908 + 0.331695i
\(232\) 0 0
\(233\) 1.46997 0.0963010 0.0481505 0.998840i \(-0.484667\pi\)
0.0481505 + 0.998840i \(0.484667\pi\)
\(234\) 0 0
\(235\) −22.6555 −1.47788
\(236\) 0 0
\(237\) −13.4013 17.2351i −0.870510 1.11954i
\(238\) 0 0
\(239\) 1.07490 + 4.01157i 0.0695293 + 0.259487i 0.991937 0.126730i \(-0.0404482\pi\)
−0.922408 + 0.386217i \(0.873782\pi\)
\(240\) 0 0
\(241\) −6.19628 + 6.19628i −0.399137 + 0.399137i −0.877929 0.478791i \(-0.841075\pi\)
0.478791 + 0.877929i \(0.341075\pi\)
\(242\) 0 0
\(243\) −12.0567 + 9.88109i −0.773438 + 0.633872i
\(244\) 0 0
\(245\) 19.2196 5.14987i 1.22789 0.329013i
\(246\) 0 0
\(247\) −15.1763 + 3.50358i −0.965648 + 0.222928i
\(248\) 0 0
\(249\) 7.63674 + 0.955636i 0.483959 + 0.0605610i
\(250\) 0 0
\(251\) −8.12469 + 14.0724i −0.512826 + 0.888240i 0.487064 + 0.873367i \(0.338068\pi\)
−0.999889 + 0.0148739i \(0.995265\pi\)
\(252\) 0 0
\(253\) 6.61142 1.77152i 0.415656 0.111375i
\(254\) 0 0
\(255\) −37.7520 4.72416i −2.36412 0.295838i
\(256\) 0 0
\(257\) 1.63406 0.101930 0.0509648 0.998700i \(-0.483770\pi\)
0.0509648 + 0.998700i \(0.483770\pi\)
\(258\) 0 0
\(259\) 0.688231 0.397350i 0.0427646 0.0246902i
\(260\) 0 0
\(261\) −12.8561 3.26873i −0.795774 0.202329i
\(262\) 0 0
\(263\) 12.9065 + 7.45155i 0.795846 + 0.459482i 0.842017 0.539452i \(-0.181368\pi\)
−0.0461705 + 0.998934i \(0.514702\pi\)
\(264\) 0 0
\(265\) −21.9146 + 5.87201i −1.34621 + 0.360715i
\(266\) 0 0
\(267\) 1.80147 14.3960i 0.110248 0.881024i
\(268\) 0 0
\(269\) −2.95682 1.70712i −0.180280 0.104085i 0.407144 0.913364i \(-0.366525\pi\)
−0.587424 + 0.809279i \(0.699858\pi\)
\(270\) 0 0
\(271\) −8.83826 2.36821i −0.536886 0.143858i −0.0198199 0.999804i \(-0.506309\pi\)
−0.517066 + 0.855945i \(0.672976\pi\)
\(272\) 0 0
\(273\) 2.64535 3.25125i 0.160104 0.196774i
\(274\) 0 0
\(275\) 14.0013 14.0013i 0.844312 0.844312i
\(276\) 0 0
\(277\) 8.61864i 0.517844i −0.965898 0.258922i \(-0.916633\pi\)
0.965898 0.258922i \(-0.0833673\pi\)
\(278\) 0 0
\(279\) −3.68786 + 2.19270i −0.220787 + 0.131274i
\(280\) 0 0
\(281\) 0.0542955 0.202633i 0.00323900 0.0120881i −0.964287 0.264858i \(-0.914675\pi\)
0.967526 + 0.252770i \(0.0813416\pi\)
\(282\) 0 0
\(283\) 23.0339i 1.36922i 0.728907 + 0.684612i \(0.240028\pi\)
−0.728907 + 0.684612i \(0.759972\pi\)
\(284\) 0 0
\(285\) −13.9532 17.9448i −0.826514 1.06296i
\(286\) 0 0
\(287\) 5.27134i 0.311157i
\(288\) 0 0
\(289\) −17.6395 30.5525i −1.03762 1.79720i
\(290\) 0 0
\(291\) 21.5373 + 16.3079i 1.26254 + 0.955986i
\(292\) 0 0
\(293\) −6.67234 + 24.9015i −0.389803 + 1.45476i 0.440653 + 0.897678i \(0.354747\pi\)
−0.830455 + 0.557085i \(0.811920\pi\)
\(294\) 0 0
\(295\) −12.5680 −0.731740
\(296\) 0 0
\(297\) 19.0109 15.1772i 1.10312 0.880669i
\(298\) 0 0
\(299\) −5.26825 0.184022i −0.304671 0.0106423i
\(300\) 0 0
\(301\) −4.49394 4.49394i −0.259027 0.259027i
\(302\) 0 0
\(303\) −6.37928 + 15.6846i −0.366480 + 0.901055i
\(304\) 0 0
\(305\) 4.41564 + 16.4794i 0.252839 + 0.943606i
\(306\) 0 0
\(307\) 23.1111 23.1111i 1.31902 1.31902i 0.404467 0.914553i \(-0.367457\pi\)
0.914553 0.404467i \(-0.132543\pi\)
\(308\) 0 0
\(309\) 18.0299 + 7.33317i 1.02568 + 0.417169i
\(310\) 0 0
\(311\) −8.87785 + 15.3769i −0.503417 + 0.871943i 0.496576 + 0.867993i \(0.334591\pi\)
−0.999992 + 0.00394966i \(0.998743\pi\)
\(312\) 0 0
\(313\) −4.89955 8.48627i −0.276939 0.479672i 0.693684 0.720280i \(-0.255987\pi\)
−0.970622 + 0.240608i \(0.922653\pi\)
\(314\) 0 0
\(315\) 5.92849 + 1.50735i 0.334033 + 0.0849293i
\(316\) 0 0
\(317\) 10.1289 + 2.71402i 0.568894 + 0.152435i 0.531789 0.846877i \(-0.321520\pi\)
0.0371046 + 0.999311i \(0.488187\pi\)
\(318\) 0 0
\(319\) 19.9952 + 5.35770i 1.11952 + 0.299974i
\(320\) 0 0
\(321\) −23.1999 + 18.0394i −1.29489 + 1.00686i
\(322\) 0 0
\(323\) 8.08407 30.1702i 0.449810 1.67871i
\(324\) 0 0
\(325\) −13.4649 + 7.15924i −0.746897 + 0.397123i
\(326\) 0 0
\(327\) −1.00435 0.760489i −0.0555409 0.0420552i
\(328\) 0 0
\(329\) 2.50259 + 4.33461i 0.137972 + 0.238975i
\(330\) 0 0
\(331\) −10.9971 10.9971i −0.604458 0.604458i 0.337034 0.941492i \(-0.390576\pi\)
−0.941492 + 0.337034i \(0.890576\pi\)
\(332\) 0 0
\(333\) −3.55185 + 0.0455386i −0.194640 + 0.00249550i
\(334\) 0 0
\(335\) −10.3629 + 17.9491i −0.566187 + 0.980664i
\(336\) 0 0
\(337\) 16.3011 + 9.41142i 0.887975 + 0.512673i 0.873280 0.487219i \(-0.161989\pi\)
0.0146957 + 0.999892i \(0.495322\pi\)
\(338\) 0 0
\(339\) 13.4535 10.4609i 0.730695 0.568160i
\(340\) 0 0
\(341\) 5.79838 3.34770i 0.314000 0.181288i
\(342\) 0 0
\(343\) −6.43049 6.43049i −0.347214 0.347214i
\(344\) 0 0
\(345\) −2.98959 7.08863i −0.160954 0.381639i
\(346\) 0 0
\(347\) 11.2620 6.50212i 0.604576 0.349052i −0.166264 0.986081i \(-0.553170\pi\)
0.770840 + 0.637029i \(0.219837\pi\)
\(348\) 0 0
\(349\) 5.09300 + 19.0073i 0.272622 + 1.01744i 0.957418 + 0.288705i \(0.0932247\pi\)
−0.684796 + 0.728735i \(0.740109\pi\)
\(350\) 0 0
\(351\) −17.4192 + 6.89720i −0.929768 + 0.368145i
\(352\) 0 0
\(353\) −0.521426 1.94599i −0.0277527 0.103575i 0.950660 0.310234i \(-0.100407\pi\)
−0.978413 + 0.206660i \(0.933741\pi\)
\(354\) 0 0
\(355\) 23.6192 13.6366i 1.25358 0.723753i
\(356\) 0 0
\(357\) 3.26633 + 7.74481i 0.172872 + 0.409899i
\(358\) 0 0
\(359\) 12.9143 + 12.9143i 0.681589 + 0.681589i 0.960358 0.278769i \(-0.0899263\pi\)
−0.278769 + 0.960358i \(0.589926\pi\)
\(360\) 0 0
\(361\) −0.293341 + 0.169361i −0.0154390 + 0.00891372i
\(362\) 0 0
\(363\) −14.9273 + 11.6069i −0.783480 + 0.609205i
\(364\) 0 0
\(365\) −31.1695 17.9957i −1.63149 0.941938i
\(366\) 0 0
\(367\) −6.46432 + 11.1965i −0.337435 + 0.584454i −0.983949 0.178447i \(-0.942893\pi\)
0.646515 + 0.762902i \(0.276226\pi\)
\(368\) 0 0
\(369\) −11.5183 + 20.5544i −0.599619 + 1.07002i
\(370\) 0 0
\(371\) 3.54422 + 3.54422i 0.184007 + 0.184007i
\(372\) 0 0
\(373\) 11.1109 + 19.2446i 0.575301 + 0.996450i 0.996009 + 0.0892538i \(0.0284482\pi\)
−0.420708 + 0.907196i \(0.638218\pi\)
\(374\) 0 0
\(375\) 3.23212 + 2.44733i 0.166906 + 0.126380i
\(376\) 0 0
\(377\) −13.5201 8.44850i −0.696322 0.435120i
\(378\) 0 0
\(379\) 7.09076 26.4631i 0.364228 1.35932i −0.504237 0.863565i \(-0.668226\pi\)
0.868465 0.495751i \(-0.165107\pi\)
\(380\) 0 0
\(381\) −19.3545 + 15.0493i −0.991561 + 0.771000i
\(382\) 0 0
\(383\) −6.64541 1.78063i −0.339564 0.0909860i 0.0850073 0.996380i \(-0.472909\pi\)
−0.424572 + 0.905394i \(0.639575\pi\)
\(384\) 0 0
\(385\) −9.22062 2.47066i −0.469926 0.125916i
\(386\) 0 0
\(387\) 7.70350 + 27.3428i 0.391591 + 1.38991i
\(388\) 0 0
\(389\) 5.10145 + 8.83597i 0.258654 + 0.448002i 0.965882 0.258984i \(-0.0833878\pi\)
−0.707228 + 0.706986i \(0.750054\pi\)
\(390\) 0 0
\(391\) 5.28559 9.15491i 0.267304 0.462983i
\(392\) 0 0
\(393\) 4.24301 + 1.72573i 0.214032 + 0.0870516i
\(394\) 0 0
\(395\) 27.0777 27.0777i 1.36243 1.36243i
\(396\) 0 0
\(397\) −5.15554 19.2407i −0.258749 0.965665i −0.965966 0.258669i \(-0.916716\pi\)
0.707217 0.706997i \(-0.249950\pi\)
\(398\) 0 0
\(399\) −1.89201 + 4.65184i −0.0947190 + 0.232883i
\(400\) 0 0
\(401\) 18.4659 + 18.4659i 0.922144 + 0.922144i 0.997181 0.0750371i \(-0.0239075\pi\)
−0.0750371 + 0.997181i \(0.523908\pi\)
\(402\) 0 0
\(403\) −5.02437 + 1.15992i −0.250282 + 0.0577796i
\(404\) 0 0
\(405\) −19.8231 18.8318i −0.985020 0.935759i
\(406\) 0 0
\(407\) 5.54319 0.274766
\(408\) 0 0
\(409\) 10.2504 38.2549i 0.506849 1.89159i 0.0572376 0.998361i \(-0.481771\pi\)
0.449611 0.893224i \(-0.351563\pi\)
\(410\) 0 0
\(411\) 0.133960 + 0.101434i 0.00660777 + 0.00500335i
\(412\) 0 0
\(413\) 1.38830 + 2.40460i 0.0683137 + 0.118323i
\(414\) 0 0
\(415\) 13.4993i 0.662655i
\(416\) 0 0
\(417\) 12.8530 + 16.5298i 0.629413 + 0.809470i
\(418\) 0 0
\(419\) 4.13333i 0.201926i −0.994890 0.100963i \(-0.967808\pi\)
0.994890 0.100963i \(-0.0321924\pi\)
\(420\) 0 0
\(421\) −0.174519 + 0.651312i −0.00850551 + 0.0317430i −0.970048 0.242913i \(-0.921897\pi\)
0.961543 + 0.274656i \(0.0885638\pi\)
\(422\) 0 0
\(423\) −0.286811 22.3702i −0.0139452 1.08768i
\(424\) 0 0
\(425\) 30.5814i 1.48341i
\(426\) 0 0
\(427\) 2.66518 2.66518i 0.128977 0.128977i
\(428\) 0 0
\(429\) 27.3188 10.4139i 1.31896 0.502786i
\(430\) 0 0
\(431\) 15.1112 + 4.04903i 0.727879 + 0.195035i 0.603685 0.797223i \(-0.293699\pi\)
0.124195 + 0.992258i \(0.460365\pi\)
\(432\) 0 0
\(433\) 32.7772 + 18.9239i 1.57517 + 0.909427i 0.995519 + 0.0945639i \(0.0301457\pi\)
0.579654 + 0.814863i \(0.303188\pi\)
\(434\) 0 0
\(435\) 2.88903 23.0870i 0.138518 1.10694i
\(436\) 0 0
\(437\) 6.10062 1.63466i 0.291832 0.0781963i
\(438\) 0 0
\(439\) −18.8319 10.8726i −0.898799 0.518922i −0.0219889 0.999758i \(-0.507000\pi\)
−0.876810 + 0.480836i \(0.840333\pi\)
\(440\) 0 0
\(441\) 5.32832 + 18.9123i 0.253730 + 0.900586i
\(442\) 0 0
\(443\) 1.49013 0.860330i 0.0707984 0.0408755i −0.464183 0.885739i \(-0.653652\pi\)
0.534981 + 0.844864i \(0.320319\pi\)
\(444\) 0 0
\(445\) 25.4476 1.20633
\(446\) 0 0
\(447\) 21.7053 + 2.71612i 1.02662 + 0.128468i
\(448\) 0 0
\(449\) −7.08109 + 1.89737i −0.334177 + 0.0895425i −0.422006 0.906593i \(-0.638674\pi\)
0.0878286 + 0.996136i \(0.472007\pi\)
\(450\) 0 0
\(451\) 18.3843 31.8425i 0.865682 1.49941i
\(452\) 0 0
\(453\) 25.7699 + 3.22476i 1.21078 + 0.151512i
\(454\) 0 0
\(455\) 6.23469 + 3.89595i 0.292287 + 0.182645i
\(456\) 0 0
\(457\) −4.08759 + 1.09527i −0.191210 + 0.0512345i −0.353153 0.935566i \(-0.614890\pi\)
0.161943 + 0.986800i \(0.448224\pi\)
\(458\) 0 0
\(459\) 4.18673 37.3363i 0.195420 1.74271i
\(460\) 0 0
\(461\) 17.1299 17.1299i 0.797821 0.797821i −0.184931 0.982752i \(-0.559206\pi\)
0.982752 + 0.184931i \(0.0592060\pi\)
\(462\) 0 0
\(463\) −1.81468 6.77247i −0.0843353 0.314744i 0.910852 0.412733i \(-0.135426\pi\)
−0.995187 + 0.0979891i \(0.968759\pi\)
\(464\) 0 0
\(465\) −4.61942 5.94090i −0.214220 0.275503i
\(466\) 0 0
\(467\) −36.8585 −1.70561 −0.852804 0.522231i \(-0.825100\pi\)
−0.852804 + 0.522231i \(0.825100\pi\)
\(468\) 0 0
\(469\) 4.57886 0.211432
\(470\) 0 0
\(471\) 1.73857 4.27459i 0.0801093 0.196963i
\(472\) 0 0
\(473\) −11.4735 42.8196i −0.527550 1.96885i
\(474\) 0 0
\(475\) 12.9196 12.9196i 0.592792 0.592792i
\(476\) 0 0
\(477\) −6.07548 21.5643i −0.278177 0.987361i
\(478\) 0 0
\(479\) 16.5400 4.43188i 0.755732 0.202498i 0.139673 0.990198i \(-0.455395\pi\)
0.616059 + 0.787700i \(0.288728\pi\)
\(480\) 0 0
\(481\) −4.08259 1.24821i −0.186150 0.0569137i
\(482\) 0 0
\(483\) −1.02601 + 1.35502i −0.0466849 + 0.0616553i
\(484\) 0 0
\(485\) −23.6921 + 41.0359i −1.07580 + 1.86334i
\(486\) 0 0
\(487\) −11.2868 + 3.02430i −0.511455 + 0.137044i −0.505311 0.862937i \(-0.668622\pi\)
−0.00614389 + 0.999981i \(0.501956\pi\)
\(488\) 0 0
\(489\) 3.15400 + 7.47847i 0.142629 + 0.338188i
\(490\) 0 0
\(491\) 22.9782 1.03699 0.518496 0.855080i \(-0.326492\pi\)
0.518496 + 0.855080i \(0.326492\pi\)
\(492\) 0 0
\(493\) 27.6876 15.9854i 1.24699 0.719948i
\(494\) 0 0
\(495\) 30.5551 + 29.7816i 1.37335 + 1.33858i
\(496\) 0 0
\(497\) −5.21807 3.01266i −0.234063 0.135136i
\(498\) 0 0
\(499\) −12.3864 + 3.31892i −0.554491 + 0.148575i −0.525174 0.850995i \(-0.676000\pi\)
−0.0293168 + 0.999570i \(0.509333\pi\)
\(500\) 0 0
\(501\) 21.5378 + 16.3082i 0.962237 + 0.728599i
\(502\) 0 0
\(503\) −1.38363 0.798839i −0.0616930 0.0356184i 0.468836 0.883285i \(-0.344673\pi\)
−0.530529 + 0.847667i \(0.678007\pi\)
\(504\) 0 0
\(505\) −28.6872 7.68671i −1.27656 0.342054i
\(506\) 0 0
\(507\) −22.4654 + 1.51821i −0.997724 + 0.0674262i
\(508\) 0 0
\(509\) −18.0128 + 18.0128i −0.798403 + 0.798403i −0.982844 0.184441i \(-0.940953\pi\)
0.184441 + 0.982844i \(0.440953\pi\)
\(510\) 0 0
\(511\) 7.95141i 0.351750i
\(512\) 0 0
\(513\) 17.5421 14.0046i 0.774503 0.618318i
\(514\) 0 0
\(515\) −8.83609 + 32.9767i −0.389365 + 1.45313i
\(516\) 0 0
\(517\) 34.9120i 1.53543i
\(518\) 0 0
\(519\) 34.3337 4.74422i 1.50708 0.208248i
\(520\) 0 0
\(521\) 24.5314i 1.07474i −0.843346 0.537370i \(-0.819418\pi\)
0.843346 0.537370i \(-0.180582\pi\)
\(522\) 0 0
\(523\) 3.69447 + 6.39900i 0.161548 + 0.279809i 0.935424 0.353528i \(-0.115018\pi\)
−0.773876 + 0.633337i \(0.781685\pi\)
\(524\) 0 0
\(525\) −0.610520 + 4.87883i −0.0266453 + 0.212929i
\(526\) 0 0
\(527\) 2.67636 9.98832i 0.116584 0.435098i
\(528\) 0 0
\(529\) −20.8624 −0.907062
\(530\) 0 0
\(531\) −0.159107 12.4098i −0.00690465 0.538538i
\(532\) 0 0
\(533\) −20.7104 + 19.3124i −0.897068 + 0.836513i
\(534\) 0 0
\(535\) −36.4490 36.4490i −1.57583 1.57583i
\(536\) 0 0
\(537\) −12.2165 15.7113i −0.527181 0.677992i
\(538\) 0 0
\(539\) −7.93591 29.6172i −0.341824 1.27570i
\(540\) 0 0
\(541\) −5.03374 + 5.03374i −0.216418 + 0.216418i −0.806987 0.590569i \(-0.798903\pi\)
0.590569 + 0.806987i \(0.298903\pi\)
\(542\) 0 0
\(543\) −6.19535 44.8355i −0.265868 1.92408i
\(544\) 0 0
\(545\) 1.10484 1.91363i 0.0473260 0.0819710i
\(546\) 0 0
\(547\) 9.69757 + 16.7967i 0.414638 + 0.718174i 0.995390 0.0959061i \(-0.0305748\pi\)
−0.580752 + 0.814080i \(0.697242\pi\)
\(548\) 0 0
\(549\) −16.2159 + 4.56864i −0.692079 + 0.194985i
\(550\) 0 0
\(551\) 18.4504 + 4.94377i 0.786013 + 0.210612i
\(552\) 0 0
\(553\) −8.17175 2.18961i −0.347498 0.0931119i
\(554\) 0 0
\(555\) −0.852817 6.17181i −0.0362000 0.261979i
\(556\) 0 0
\(557\) −10.5583 + 39.4040i −0.447368 + 1.66960i 0.262238 + 0.965003i \(0.415540\pi\)
−0.709606 + 0.704598i \(0.751127\pi\)
\(558\) 0 0
\(559\) −1.19184 + 34.1204i −0.0504095 + 1.44314i
\(560\) 0 0
\(561\) −7.27990 + 58.1756i −0.307357 + 2.45618i
\(562\) 0 0
\(563\) 12.5011 + 21.6525i 0.526858 + 0.912544i 0.999510 + 0.0312954i \(0.00996326\pi\)
−0.472652 + 0.881249i \(0.656703\pi\)
\(564\) 0 0
\(565\) 21.1365 + 21.1365i 0.889221 + 0.889221i
\(566\) 0 0
\(567\) −1.41331 + 5.87290i −0.0593534 + 0.246639i
\(568\) 0 0
\(569\) −5.30226 + 9.18378i −0.222282 + 0.385004i −0.955501 0.294989i \(-0.904684\pi\)
0.733218 + 0.679993i \(0.238017\pi\)
\(570\) 0 0
\(571\) 11.4489 + 6.61000i 0.479120 + 0.276620i 0.720050 0.693923i \(-0.244119\pi\)
−0.240930 + 0.970543i \(0.577452\pi\)
\(572\) 0 0
\(573\) −0.564023 4.08182i −0.0235624 0.170520i
\(574\) 0 0
\(575\) 5.35530 3.09189i 0.223332 0.128941i
\(576\) 0 0
\(577\) 17.9959 + 17.9959i 0.749178 + 0.749178i 0.974325 0.225147i \(-0.0722861\pi\)
−0.225147 + 0.974325i \(0.572286\pi\)
\(578\) 0 0
\(579\) 17.3631 22.9310i 0.721587 0.952978i
\(580\) 0 0
\(581\) 2.58278 1.49117i 0.107152 0.0618641i
\(582\) 0 0
\(583\) 9.04872 + 33.7703i 0.374760 + 1.39862i
\(584\) 0 0
\(585\) −15.7978 28.8147i −0.653159 1.19134i
\(586\) 0 0
\(587\) −5.52782 20.6301i −0.228158 0.851496i −0.981115 0.193426i \(-0.938040\pi\)
0.752957 0.658069i \(-0.228627\pi\)
\(588\) 0 0
\(589\) 5.35040 3.08906i 0.220460 0.127282i
\(590\) 0 0
\(591\) −42.0023 5.25602i −1.72774 0.216204i
\(592\) 0 0
\(593\) 22.0256 + 22.0256i 0.904484 + 0.904484i 0.995820 0.0913365i \(-0.0291139\pi\)
−0.0913365 + 0.995820i \(0.529114\pi\)
\(594\) 0 0
\(595\) −12.7679 + 7.37155i −0.523433 + 0.302204i
\(596\) 0 0
\(597\) 22.0107 + 8.95228i 0.900840 + 0.366393i
\(598\) 0 0
\(599\) 14.0092 + 8.08823i 0.572401 + 0.330476i 0.758108 0.652129i \(-0.226124\pi\)
−0.185707 + 0.982605i \(0.559457\pi\)
\(600\) 0 0
\(601\) 3.07380 5.32398i 0.125383 0.217170i −0.796500 0.604639i \(-0.793317\pi\)
0.921883 + 0.387469i \(0.126651\pi\)
\(602\) 0 0
\(603\) −17.8542 10.0052i −0.727080 0.407442i
\(604\) 0 0
\(605\) −23.4520 23.4520i −0.953459 0.953459i
\(606\) 0 0
\(607\) 0.0178799 + 0.0309689i 0.000725723 + 0.00125699i 0.866388 0.499371i \(-0.166436\pi\)
−0.865662 + 0.500628i \(0.833102\pi\)
\(608\) 0 0
\(609\) −4.73630 + 1.99750i −0.191924 + 0.0809429i
\(610\) 0 0
\(611\) 7.86148 25.7129i 0.318042 1.04023i
\(612\) 0 0
\(613\) −2.11344 + 7.88746i −0.0853610 + 0.318572i −0.995382 0.0959899i \(-0.969398\pi\)
0.910021 + 0.414561i \(0.136065\pi\)
\(614\) 0 0
\(615\) −38.2820 15.5702i −1.54368 0.627851i
\(616\) 0 0
\(617\) 18.5850 + 4.97983i 0.748203 + 0.200480i 0.612721 0.790299i \(-0.290075\pi\)
0.135482 + 0.990780i \(0.456742\pi\)
\(618\) 0 0
\(619\) −17.6220 4.72181i −0.708290 0.189786i −0.113349 0.993555i \(-0.536158\pi\)
−0.594941 + 0.803770i \(0.702824\pi\)
\(620\) 0 0
\(621\) 6.96150 3.04167i 0.279356 0.122058i
\(622\) 0 0
\(623\) −2.81101 4.86881i −0.112621 0.195065i
\(624\) 0 0
\(625\) −14.1293 + 24.4727i −0.565174 + 0.978909i
\(626\) 0 0
\(627\) −27.6528 + 21.5017i −1.10434 + 0.858696i
\(628\) 0 0
\(629\) 6.05365 6.05365i 0.241375 0.241375i
\(630\) 0 0
\(631\) 8.96498 + 33.4578i 0.356890 + 1.33193i 0.878089 + 0.478497i \(0.158818\pi\)
−0.521199 + 0.853435i \(0.674515\pi\)
\(632\) 0 0
\(633\) 25.2767 3.49272i 1.00466 0.138823i
\(634\) 0 0
\(635\) −30.4074 30.4074i −1.20668 1.20668i
\(636\) 0 0
\(637\) −0.824367 + 23.6003i −0.0326626 + 0.935076i
\(638\) 0 0
\(639\) 13.7638 + 23.1491i 0.544488 + 0.915764i
\(640\) 0 0
\(641\) −3.60735 −0.142482 −0.0712409 0.997459i \(-0.522696\pi\)
−0.0712409 + 0.997459i \(0.522696\pi\)
\(642\) 0 0
\(643\) −3.92452 + 14.6465i −0.154768 + 0.577601i 0.844357 + 0.535781i \(0.179983\pi\)
−0.999125 + 0.0418209i \(0.986684\pi\)
\(644\) 0 0
\(645\) −45.9103 + 19.3624i −1.80772 + 0.762393i
\(646\) 0 0
\(647\) −23.1852 40.1580i −0.911505 1.57877i −0.811939 0.583743i \(-0.801588\pi\)
−0.0995668 0.995031i \(-0.531746\pi\)
\(648\) 0 0
\(649\) 19.3673i 0.760232i
\(650\) 0 0
\(651\) −0.626380 + 1.54006i −0.0245498 + 0.0603599i
\(652\) 0 0
\(653\) 10.8522i 0.424679i 0.977196 + 0.212340i \(0.0681083\pi\)
−0.977196 + 0.212340i \(0.931892\pi\)
\(654\) 0 0
\(655\) −2.07942 + 7.76049i −0.0812496 + 0.303228i
\(656\) 0 0
\(657\) 17.3745 31.0047i 0.677842 1.20961i
\(658\) 0 0
\(659\) 21.8367i 0.850638i 0.905043 + 0.425319i \(0.139838\pi\)
−0.905043 + 0.425319i \(0.860162\pi\)
\(660\) 0 0
\(661\) −6.77396 + 6.77396i −0.263477 + 0.263477i −0.826465 0.562988i \(-0.809652\pi\)
0.562988 + 0.826465i \(0.309652\pi\)
\(662\) 0 0
\(663\) 18.4617 41.2073i 0.716992 1.60036i
\(664\) 0 0
\(665\) −8.50824 2.27978i −0.329935 0.0884059i
\(666\) 0 0
\(667\) 5.59863 + 3.23237i 0.216780 + 0.125158i
\(668\) 0 0
\(669\) −42.2450 + 17.8165i −1.63328 + 0.688828i
\(670\) 0 0
\(671\) 25.3946 6.80446i 0.980348 0.262683i
\(672\) 0 0
\(673\) −32.6123 18.8287i −1.25711 0.725793i −0.284599 0.958647i \(-0.591860\pi\)
−0.972512 + 0.232853i \(0.925194\pi\)
\(674\) 0 0
\(675\) 13.0412 17.6899i 0.501956 0.680883i
\(676\) 0 0
\(677\) −20.6548 + 11.9251i −0.793829 + 0.458317i −0.841309 0.540555i \(-0.818214\pi\)
0.0474798 + 0.998872i \(0.484881\pi\)
\(678\) 0 0
\(679\) 10.4683 0.401738
\(680\) 0 0
\(681\) −3.81220 + 5.03465i −0.146084 + 0.192928i
\(682\) 0 0
\(683\) −37.2984 + 9.99406i −1.42718 + 0.382412i −0.888026 0.459794i \(-0.847923\pi\)
−0.539156 + 0.842206i \(0.681257\pi\)
\(684\) 0 0
\(685\) −0.147362 + 0.255239i −0.00563043 + 0.00975219i
\(686\) 0 0
\(687\) 8.29462 + 19.6674i 0.316459 + 0.750360i
\(688\) 0 0
\(689\) 0.939964 26.9096i 0.0358098 1.02517i
\(690\) 0 0
\(691\) −36.4597 + 9.76934i −1.38699 + 0.371643i −0.873656 0.486545i \(-0.838257\pi\)
−0.513335 + 0.858188i \(0.671590\pi\)
\(692\) 0 0
\(693\) 2.32281 9.13576i 0.0882362 0.347039i
\(694\) 0 0
\(695\) −25.9697 + 25.9697i −0.985087 + 0.985087i
\(696\) 0 0
\(697\) −14.6976 54.8521i −0.556710 2.07767i
\(698\) 0 0
\(699\) −2.52210 + 0.348502i −0.0953946 + 0.0131816i
\(700\) 0 0
\(701\) −47.2060 −1.78295 −0.891473 0.453073i \(-0.850328\pi\)
−0.891473 + 0.453073i \(0.850328\pi\)
\(702\) 0 0
\(703\) 5.11493 0.192913
\(704\) 0 0
\(705\) 38.8712 5.37119i 1.46397 0.202291i
\(706\) 0 0
\(707\) 1.69819 + 6.33772i 0.0638669 + 0.238354i
\(708\) 0 0
\(709\) 20.6959 20.6959i 0.777251 0.777251i −0.202112 0.979362i \(-0.564780\pi\)
0.979362 + 0.202112i \(0.0647804\pi\)
\(710\) 0 0
\(711\) 27.0794 + 26.3938i 1.01556 + 0.989846i
\(712\) 0 0
\(713\) 2.01971 0.541179i 0.0756387 0.0202673i
\(714\) 0 0
\(715\) 24.0743 + 45.2783i 0.900329 + 1.69331i
\(716\) 0 0
\(717\) −2.79532 6.62801i −0.104393 0.247527i
\(718\) 0 0
\(719\) −14.4340 + 25.0004i −0.538297 + 0.932357i 0.460699 + 0.887556i \(0.347599\pi\)
−0.998996 + 0.0448010i \(0.985735\pi\)
\(720\) 0 0
\(721\) 7.28539 1.95211i 0.271322 0.0727005i
\(722\) 0 0
\(723\) 9.16223 12.1003i 0.340747 0.450014i
\(724\) 0 0
\(725\) 18.7019 0.694570
\(726\) 0 0
\(727\) 33.1912 19.1629i 1.23099 0.710714i 0.263755 0.964590i \(-0.415039\pi\)
0.967237 + 0.253876i \(0.0817056\pi\)
\(728\) 0 0
\(729\) 18.3436 19.8119i 0.679394 0.733773i
\(730\) 0 0
\(731\) −59.2927 34.2327i −2.19302 1.26614i
\(732\) 0 0
\(733\) −38.9931 + 10.4482i −1.44024 + 0.385911i −0.892618 0.450814i \(-0.851134\pi\)
−0.547623 + 0.836725i \(0.684467\pi\)
\(734\) 0 0
\(735\) −31.7550 + 13.3925i −1.17130 + 0.493989i
\(736\) 0 0
\(737\) 27.6594 + 15.9692i 1.01885 + 0.588232i
\(738\) 0 0
\(739\) −6.38584 1.71108i −0.234907 0.0629431i 0.139445 0.990230i \(-0.455468\pi\)
−0.374352 + 0.927287i \(0.622135\pi\)
\(740\) 0 0
\(741\) 25.2082 9.60929i 0.926045 0.353006i
\(742\) 0 0
\(743\) 12.0339 12.0339i 0.441479 0.441479i −0.451030 0.892509i \(-0.648943\pi\)
0.892509 + 0.451030i \(0.148943\pi\)
\(744\) 0 0
\(745\) 38.3680i 1.40569i
\(746\) 0 0
\(747\) −13.3293 + 0.170896i −0.487693 + 0.00625277i
\(748\) 0 0
\(749\) −2.94742 + 10.9999i −0.107696 + 0.401928i
\(750\) 0 0
\(751\) 0.0965284i 0.00352237i −0.999998 0.00176119i \(-0.999439\pi\)
0.999998 0.00176119i \(-0.000560603\pi\)
\(752\) 0 0
\(753\) 10.6036 26.0709i 0.386418 0.950075i
\(754\) 0 0
\(755\) 45.5529i 1.65784i
\(756\) 0 0
\(757\) −14.1700 24.5432i −0.515019 0.892039i −0.999848 0.0174302i \(-0.994452\pi\)
0.484829 0.874609i \(-0.338882\pi\)
\(758\) 0 0
\(759\) −10.9235 + 4.60693i −0.396499 + 0.167221i
\(760\) 0 0
\(761\) −3.66031 + 13.6605i −0.132686 + 0.495192i −0.999997 0.00257596i \(-0.999180\pi\)
0.867311 + 0.497767i \(0.165847\pi\)
\(762\) 0 0
\(763\) −0.488172 −0.0176730
\(764\) 0 0
\(765\) 65.8930 0.844821i 2.38237 0.0305446i
\(766\) 0 0
\(767\) 4.36112 14.2641i 0.157471 0.515047i
\(768\) 0 0
\(769\) 3.28647 + 3.28647i 0.118513 + 0.118513i 0.763876 0.645363i \(-0.223294\pi\)
−0.645363 + 0.763876i \(0.723294\pi\)
\(770\) 0 0
\(771\) −2.80363 + 0.387404i −0.100970 + 0.0139520i
\(772\) 0 0
\(773\) −8.37392 31.2519i −0.301189 1.12405i −0.936177 0.351530i \(-0.885662\pi\)
0.634988 0.772522i \(-0.281005\pi\)
\(774\) 0 0
\(775\) 4.27724 4.27724i 0.153643 0.153643i
\(776\) 0 0
\(777\) −1.08663 + 0.844920i −0.0389825 + 0.0303113i
\(778\) 0 0
\(779\) 16.9639 29.3824i 0.607796 1.05273i
\(780\) 0 0
\(781\) −21.0138 36.3970i −0.751934 1.30239i
\(782\) 0 0
\(783\) 22.8328 + 2.56037i 0.815979 + 0.0915002i
\(784\) 0 0
\(785\) 7.81825 + 2.09489i 0.279045 + 0.0747700i
\(786\) 0 0
\(787\) 21.7742 + 5.83439i 0.776167 + 0.207973i 0.625094 0.780549i \(-0.285060\pi\)
0.151073 + 0.988523i \(0.451727\pi\)
\(788\) 0 0
\(789\) −23.9109 9.72510i −0.851249 0.346223i
\(790\) 0 0
\(791\) 1.70919 6.37878i 0.0607718 0.226803i
\(792\) 0 0
\(793\) −20.2355 0.706835i −0.718583 0.0251004i
\(794\) 0 0
\(795\) 36.2079 15.2704i 1.28416 0.541586i
\(796\) 0 0
\(797\) 3.95600 + 6.85199i 0.140129 + 0.242710i 0.927545 0.373712i \(-0.121915\pi\)
−0.787416 + 0.616422i \(0.788582\pi\)
\(798\) 0 0
\(799\) 38.1270 + 38.1270i 1.34883 + 1.34883i
\(800\) 0 0
\(801\) 0.322157 + 25.1271i 0.0113829 + 0.887822i
\(802\) 0 0
\(803\) −27.7313 + 48.0320i −0.978615 + 1.69501i
\(804\) 0 0
\(805\) −2.58176 1.49058i −0.0909951 0.0525360i
\(806\) 0 0
\(807\) 5.47788 + 2.22798i 0.192830 + 0.0784286i
\(808\) 0 0
\(809\) −18.5652 + 10.7186i −0.652716 + 0.376846i −0.789496 0.613756i \(-0.789658\pi\)
0.136780 + 0.990601i \(0.456325\pi\)
\(810\) 0 0
\(811\) 22.3538 + 22.3538i 0.784949 + 0.784949i 0.980661 0.195712i \(-0.0627019\pi\)
−0.195712 + 0.980661i \(0.562702\pi\)
\(812\) 0 0
\(813\) 15.7257 + 1.96786i 0.551524 + 0.0690158i
\(814\) 0 0
\(815\) −12.3288 + 7.11804i −0.431859 + 0.249334i
\(816\) 0 0
\(817\) −10.5870 39.5114i −0.370394 1.38233i
\(818\) 0 0
\(819\) −3.76795 + 6.20549i −0.131663 + 0.216837i
\(820\) 0 0
\(821\) 9.19464 + 34.3149i 0.320895 + 1.19760i 0.918374 + 0.395713i \(0.129502\pi\)
−0.597479 + 0.801884i \(0.703831\pi\)
\(822\) 0 0
\(823\) 29.1783 16.8461i 1.01709 0.587218i 0.103831 0.994595i \(-0.466890\pi\)
0.913260 + 0.407377i \(0.133557\pi\)
\(824\) 0 0
\(825\) −20.7033 + 27.3422i −0.720797 + 0.951934i
\(826\) 0 0
\(827\) 3.95015 + 3.95015i 0.137360 + 0.137360i 0.772444 0.635083i \(-0.219034\pi\)
−0.635083 + 0.772444i \(0.719034\pi\)
\(828\) 0 0
\(829\) 24.2862 14.0217i 0.843495 0.486992i −0.0149557 0.999888i \(-0.504761\pi\)
0.858451 + 0.512896i \(0.171427\pi\)
\(830\) 0 0
\(831\) 2.04332 + 14.7874i 0.0708819 + 0.512970i
\(832\) 0 0
\(833\) −41.0113 23.6779i −1.42096 0.820391i
\(834\) 0 0
\(835\) −23.6926 + 41.0367i −0.819915 + 1.42013i
\(836\) 0 0
\(837\) 5.80760 4.63645i 0.200740 0.160259i
\(838\) 0 0
\(839\) −25.2371 25.2371i −0.871281 0.871281i 0.121331 0.992612i \(-0.461284\pi\)
−0.992612 + 0.121331i \(0.961284\pi\)
\(840\) 0 0
\(841\) −4.72419 8.18255i −0.162903 0.282157i
\(842\) 0 0
\(843\) −0.0451168 + 0.360540i −0.00155391 + 0.0124177i
\(844\) 0 0
\(845\) −7.53510 38.7687i −0.259215 1.33369i
\(846\) 0 0
\(847\) −1.89643 + 7.07756i −0.0651620 + 0.243188i
\(848\) 0 0
\(849\) −5.46091 39.5204i −0.187418 1.35634i
\(850\) 0 0
\(851\) 1.67214 + 0.448048i 0.0573202 + 0.0153589i
\(852\) 0 0
\(853\) 21.4464 + 5.74656i 0.734312 + 0.196758i 0.606549 0.795046i \(-0.292553\pi\)
0.127763 + 0.991805i \(0.459220\pi\)
\(854\) 0 0
\(855\) 28.1945 + 27.4807i 0.964231 + 0.939819i
\(856\) 0 0
\(857\) −9.54760 16.5369i −0.326140 0.564891i 0.655603 0.755106i \(-0.272415\pi\)
−0.981742 + 0.190215i \(0.939081\pi\)
\(858\) 0 0
\(859\) −27.1199 + 46.9730i −0.925319 + 1.60270i −0.134272 + 0.990945i \(0.542870\pi\)
−0.791047 + 0.611755i \(0.790464\pi\)
\(860\) 0 0
\(861\) 1.24974 + 9.04429i 0.0425909 + 0.308229i
\(862\) 0 0
\(863\) −10.0822 + 10.0822i −0.343201 + 0.343201i −0.857569 0.514368i \(-0.828026\pi\)
0.514368 + 0.857569i \(0.328026\pi\)
\(864\) 0 0
\(865\) 15.7345 + 58.7221i 0.534990 + 1.99661i
\(866\) 0 0
\(867\) 37.5083 + 48.2383i 1.27385 + 1.63826i
\(868\) 0 0
\(869\) −41.7265 41.7265i −1.41547 1.41547i
\(870\) 0 0
\(871\) −16.7754 17.9897i −0.568412 0.609559i
\(872\) 0 0
\(873\) −40.8190 22.8742i −1.38151 0.774173i
\(874\) 0 0
\(875\) 1.57099 0.0531091
\(876\) 0 0
\(877\) 0.909826 3.39552i 0.0307227 0.114658i −0.948862 0.315692i \(-0.897763\pi\)
0.979584 + 0.201034i \(0.0644300\pi\)
\(878\) 0 0
\(879\) 5.54438 44.3066i 0.187007 1.49443i
\(880\) 0 0
\(881\) 14.0958 + 24.4146i 0.474899 + 0.822549i 0.999587 0.0287456i \(-0.00915126\pi\)
−0.524688 + 0.851295i \(0.675818\pi\)
\(882\) 0 0
\(883\) 8.66822i 0.291709i −0.989306 0.145854i \(-0.953407\pi\)
0.989306 0.145854i \(-0.0465931\pi\)
\(884\) 0 0
\(885\) 21.5636 2.97965i 0.724853 0.100160i
\(886\) 0 0
\(887\) 10.0296i 0.336760i −0.985722 0.168380i \(-0.946146\pi\)
0.985722 0.168380i \(-0.0538536\pi\)
\(888\) 0 0
\(889\) −2.45887 + 9.17664i −0.0824680 + 0.307775i
\(890\) 0 0
\(891\) −29.0196 + 30.5473i −0.972195 + 1.02337i
\(892\) 0 0
\(893\) 32.2147i 1.07802i
\(894\) 0 0
\(895\) 24.6837 24.6837i 0.825084 0.825084i
\(896\) 0 0
\(897\) 9.08263 0.933267i 0.303260 0.0311609i
\(898\) 0 0
\(899\) 6.10830 + 1.63671i 0.203723 + 0.0545875i
\(900\) 0 0
\(901\) 46.7621 + 26.9981i 1.55787 + 0.899438i
\(902\) 0 0
\(903\) 8.77591 + 6.64505i 0.292044 + 0.221133i
\(904\) 0 0
\(905\) 76.6836 20.5473i 2.54905 0.683016i
\(906\) 0 0
\(907\) −23.2795 13.4404i −0.772983 0.446282i 0.0609549 0.998141i \(-0.480585\pi\)
−0.833938 + 0.551859i \(0.813919\pi\)
\(908\) 0 0
\(909\) 7.22673 28.4232i 0.239695 0.942738i
\(910\) 0 0
\(911\) 37.9403 21.9048i 1.25702 0.725740i 0.284525 0.958669i \(-0.408164\pi\)
0.972494 + 0.232929i \(0.0748309\pi\)
\(912\) 0 0
\(913\) 20.8024 0.688457
\(914\) 0 0
\(915\) −11.4831 27.2276i −0.379619 0.900117i
\(916\) 0 0
\(917\) 1.71449 0.459395i 0.0566174 0.0151706i
\(918\) 0 0
\(919\) −0.847859 + 1.46853i −0.0279683 + 0.0484425i −0.879671 0.475583i \(-0.842237\pi\)
0.851702 + 0.524026i \(0.175570\pi\)
\(920\) 0 0
\(921\) −34.1736 + 45.1320i −1.12606 + 1.48715i
\(922\) 0 0
\(923\) 7.28091 + 31.5385i 0.239654 + 1.03810i
\(924\) 0 0
\(925\) 4.83733 1.29616i 0.159051 0.0426175i
\(926\) 0 0
\(927\) −32.6733 8.30733i −1.07313 0.272849i
\(928\) 0 0
\(929\) 36.1283 36.1283i 1.18533 1.18533i 0.206987 0.978344i \(-0.433634\pi\)
0.978344 0.206987i \(-0.0663658\pi\)
\(930\) 0 0
\(931\) −7.32279 27.3290i −0.239995 0.895673i
\(932\) 0 0
\(933\) 11.5866 28.4876i 0.379328 0.932643i
\(934\) 0 0
\(935\) −102.836 −3.36309
\(936\) 0 0
\(937\) −7.45635 −0.243588 −0.121794 0.992555i \(-0.538865\pi\)
−0.121794 + 0.992555i \(0.538865\pi\)
\(938\) 0 0
\(939\) 10.4183 + 13.3987i 0.339989 + 0.437251i
\(940\) 0 0
\(941\) −7.11870 26.5673i −0.232063 0.866070i −0.979451 0.201683i \(-0.935359\pi\)
0.747388 0.664388i \(-0.231308\pi\)
\(942\) 0 0
\(943\) 8.11953 8.11953i 0.264408 0.264408i
\(944\) 0 0
\(945\) −10.5292 1.18069i −0.342514 0.0384080i
\(946\) 0 0
\(947\) −24.9829 + 6.69414i −0.811834 + 0.217530i −0.640773 0.767730i \(-0.721386\pi\)
−0.171061 + 0.985260i \(0.554719\pi\)
\(948\) 0 0
\(949\) 31.2401 29.1313i 1.01410 0.945641i
\(950\) 0 0
\(951\) −18.0220 2.25522i −0.584404 0.0731304i
\(952\) 0 0
\(953\) 23.8503 41.3099i 0.772586 1.33816i −0.163555 0.986534i \(-0.552296\pi\)
0.936141 0.351624i \(-0.114371\pi\)
\(954\) 0 0
\(955\) 6.98126 1.87062i 0.225908 0.0605319i
\(956\) 0 0
\(957\) −35.5770 4.45198i −1.15004 0.143912i
\(958\) 0 0
\(959\) 0.0651121 0.00210258
\(960\) 0 0
\(961\) −25.0755 + 14.4773i −0.808886 + 0.467010i
\(962\) 0 0
\(963\) 35.5285 36.4513i 1.14489 1.17463i
\(964\) 0 0
\(965\) 43.6912 + 25.2251i 1.40647 + 0.812025i
\(966\) 0 0
\(967\) −53.1577 + 14.2436i −1.70944 + 0.458042i −0.975286 0.220944i \(-0.929086\pi\)
−0.734151 + 0.678986i \(0.762419\pi\)
\(968\) 0 0
\(969\) −6.71746 + 53.6810i −0.215796 + 1.72448i
\(970\) 0 0
\(971\) −22.0094 12.7071i −0.706316 0.407792i 0.103379 0.994642i \(-0.467034\pi\)
−0.809696 + 0.586850i \(0.800368\pi\)
\(972\) 0 0
\(973\) 7.83738 + 2.10002i 0.251255 + 0.0673235i
\(974\) 0 0
\(975\) 21.4050 15.4757i 0.685509 0.495620i
\(976\) 0 0
\(977\) 31.1030 31.1030i 0.995074 0.995074i −0.00491404 0.999988i \(-0.501564\pi\)
0.999988 + 0.00491404i \(0.00156419\pi\)
\(978\) 0 0
\(979\) 39.2146i 1.25330i
\(980\) 0 0
\(981\) 1.90352 + 1.06670i 0.0607746 + 0.0340570i
\(982\) 0 0
\(983\) −11.9175 + 44.4766i −0.380108 + 1.41858i 0.465627 + 0.884981i \(0.345829\pi\)
−0.845735 + 0.533603i \(0.820838\pi\)
\(984\) 0 0
\(985\) 74.2466i 2.36569i
\(986\) 0 0
\(987\) −5.32146 6.84378i −0.169384 0.217840i
\(988\) 0 0
\(989\) 13.8442i 0.440219i
\(990\) 0 0
\(991\) −16.6829 28.8957i −0.529951 0.917901i −0.999390 0.0349364i \(-0.988877\pi\)
0.469439 0.882965i \(-0.344456\pi\)
\(992\) 0 0
\(993\) 21.4756 + 16.2611i 0.681506 + 0.516031i
\(994\) 0 0
\(995\) −10.7870 + 40.2578i −0.341972 + 1.27626i
\(996\) 0 0
\(997\) 17.5089 0.554512 0.277256 0.960796i \(-0.410575\pi\)
0.277256 + 0.960796i \(0.410575\pi\)
\(998\) 0 0
\(999\) 6.08328 0.920208i 0.192466 0.0291141i
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 468.2.bz.a.401.1 yes 56
3.2 odd 2 1404.2.cc.a.557.13 56
9.2 odd 6 468.2.bw.a.245.11 yes 56
9.7 even 3 1404.2.bz.a.89.13 56
13.6 odd 12 468.2.bw.a.149.11 56
39.32 even 12 1404.2.bz.a.773.13 56
117.97 odd 12 1404.2.cc.a.305.13 56
117.110 even 12 inner 468.2.bz.a.461.1 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
468.2.bw.a.149.11 56 13.6 odd 12
468.2.bw.a.245.11 yes 56 9.2 odd 6
468.2.bz.a.401.1 yes 56 1.1 even 1 trivial
468.2.bz.a.461.1 yes 56 117.110 even 12 inner
1404.2.bz.a.89.13 56 9.7 even 3
1404.2.bz.a.773.13 56 39.32 even 12
1404.2.cc.a.305.13 56 117.97 odd 12
1404.2.cc.a.557.13 56 3.2 odd 2