Properties

Label 324.4.i.a.73.6
Level $324$
Weight $4$
Character 324.73
Analytic conductor $19.117$
Analytic rank $0$
Dimension $54$
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] = [324,4,Mod(37,324)]
 
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, 14]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.i (of order \(9\), 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: \(19.1166188419\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 73.6
Character \(\chi\) \(=\) 324.73
Dual form 324.4.i.a.253.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.18194 - 4.34817i) q^{5} +(16.8240 - 6.12342i) q^{7} +O(q^{10})\) \(q+(5.18194 - 4.34817i) q^{5} +(16.8240 - 6.12342i) q^{7} +(-45.5667 - 38.2350i) q^{11} +(-5.02544 - 28.5007i) q^{13} +(14.6698 + 25.4089i) q^{17} +(13.7884 - 23.8822i) q^{19} +(0.946050 + 0.344334i) q^{23} +(-13.7600 + 78.0370i) q^{25} +(15.4827 - 87.8069i) q^{29} +(-11.6374 - 4.23566i) q^{31} +(60.5552 - 104.885i) q^{35} +(-186.599 - 323.199i) q^{37} +(-38.3921 - 217.733i) q^{41} +(-415.169 - 348.368i) q^{43} +(575.029 - 209.294i) q^{47} +(-17.2036 + 14.4355i) q^{49} +51.1253 q^{53} -402.376 q^{55} +(326.874 - 274.280i) q^{59} +(488.734 - 177.884i) q^{61} +(-149.967 - 125.838i) q^{65} +(38.0132 + 215.584i) q^{67} +(99.9846 + 173.178i) q^{71} +(-398.711 + 690.588i) q^{73} +(-1000.74 - 364.240i) q^{77} +(227.171 - 1288.35i) q^{79} +(11.1689 - 63.3422i) q^{83} +(186.500 + 67.8805i) q^{85} +(-466.341 + 807.726i) q^{89} +(-259.070 - 448.722i) q^{91} +(-32.3931 - 183.711i) q^{95} +(915.498 + 768.194i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 54 q - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{9}\right)\)

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
\(5\) 5.18194 4.34817i 0.463487 0.388912i −0.380925 0.924606i \(-0.624394\pi\)
0.844412 + 0.535694i \(0.179950\pi\)
\(6\) 0 0
\(7\) 16.8240 6.12342i 0.908409 0.330634i 0.154792 0.987947i \(-0.450529\pi\)
0.753617 + 0.657313i \(0.228307\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −45.5667 38.2350i −1.24899 1.04803i −0.996766 0.0803571i \(-0.974394\pi\)
−0.252223 0.967669i \(-0.581162\pi\)
\(12\) 0 0
\(13\) −5.02544 28.5007i −0.107216 0.608052i −0.990312 0.138860i \(-0.955656\pi\)
0.883096 0.469192i \(-0.155455\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.6698 + 25.4089i 0.209291 + 0.362503i 0.951491 0.307675i \(-0.0995510\pi\)
−0.742200 + 0.670178i \(0.766218\pi\)
\(18\) 0 0
\(19\) 13.7884 23.8822i 0.166488 0.288366i −0.770695 0.637205i \(-0.780091\pi\)
0.937183 + 0.348839i \(0.113424\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.946050 + 0.344334i 0.00857674 + 0.00312168i 0.346305 0.938122i \(-0.387436\pi\)
−0.337728 + 0.941244i \(0.609658\pi\)
\(24\) 0 0
\(25\) −13.7600 + 78.0370i −0.110080 + 0.624296i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 15.4827 87.8069i 0.0991404 0.562253i −0.894259 0.447549i \(-0.852297\pi\)
0.993400 0.114704i \(-0.0365919\pi\)
\(30\) 0 0
\(31\) −11.6374 4.23566i −0.0674238 0.0245402i 0.308088 0.951358i \(-0.400311\pi\)
−0.375512 + 0.926818i \(0.622533\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 60.5552 104.885i 0.292448 0.506536i
\(36\) 0 0
\(37\) −186.599 323.199i −0.829100 1.43604i −0.898745 0.438472i \(-0.855520\pi\)
0.0696442 0.997572i \(-0.477814\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −38.3921 217.733i −0.146240 0.829369i −0.966363 0.257182i \(-0.917206\pi\)
0.820123 0.572187i \(-0.193905\pi\)
\(42\) 0 0
\(43\) −415.169 348.368i −1.47239 1.23548i −0.913882 0.405979i \(-0.866931\pi\)
−0.558505 0.829501i \(-0.688625\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 575.029 209.294i 1.78461 0.649545i 0.785063 0.619416i \(-0.212631\pi\)
0.999546 0.0301286i \(-0.00959170\pi\)
\(48\) 0 0
\(49\) −17.2036 + 14.4355i −0.0501562 + 0.0420861i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 51.1253 0.132502 0.0662509 0.997803i \(-0.478896\pi\)
0.0662509 + 0.997803i \(0.478896\pi\)
\(54\) 0 0
\(55\) −402.376 −0.986480
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 326.874 274.280i 0.721278 0.605224i −0.206460 0.978455i \(-0.566194\pi\)
0.927739 + 0.373231i \(0.121750\pi\)
\(60\) 0 0
\(61\) 488.734 177.884i 1.02583 0.373373i 0.226341 0.974048i \(-0.427324\pi\)
0.799493 + 0.600675i \(0.205101\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −149.967 125.838i −0.286172 0.240127i
\(66\) 0 0
\(67\) 38.0132 + 215.584i 0.0693143 + 0.393101i 0.999652 + 0.0263958i \(0.00840303\pi\)
−0.930337 + 0.366705i \(0.880486\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 99.9846 + 173.178i 0.167127 + 0.289472i 0.937408 0.348232i \(-0.113218\pi\)
−0.770282 + 0.637704i \(0.779884\pi\)
\(72\) 0 0
\(73\) −398.711 + 690.588i −0.639255 + 1.10722i 0.346342 + 0.938108i \(0.387424\pi\)
−0.985597 + 0.169113i \(0.945910\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1000.74 364.240i −1.48111 0.539078i
\(78\) 0 0
\(79\) 227.171 1288.35i 0.323529 1.83482i −0.196290 0.980546i \(-0.562889\pi\)
0.519819 0.854277i \(-0.326000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.1689 63.3422i 0.0147705 0.0837675i −0.976531 0.215375i \(-0.930903\pi\)
0.991302 + 0.131607i \(0.0420138\pi\)
\(84\) 0 0
\(85\) 186.500 + 67.8805i 0.237986 + 0.0866197i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −466.341 + 807.726i −0.555416 + 0.962009i 0.442455 + 0.896791i \(0.354108\pi\)
−0.997871 + 0.0652184i \(0.979226\pi\)
\(90\) 0 0
\(91\) −259.070 448.722i −0.298439 0.516911i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −32.3931 183.711i −0.0349839 0.198403i
\(96\) 0 0
\(97\) 915.498 + 768.194i 0.958296 + 0.804106i 0.980675 0.195644i \(-0.0626797\pi\)
−0.0223792 + 0.999750i \(0.507124\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 934.108 339.987i 0.920269 0.334951i 0.161924 0.986803i \(-0.448230\pi\)
0.758346 + 0.651853i \(0.226008\pi\)
\(102\) 0 0
\(103\) −252.152 + 211.581i −0.241216 + 0.202405i −0.755379 0.655288i \(-0.772547\pi\)
0.514163 + 0.857693i \(0.328103\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 321.330 0.290319 0.145159 0.989408i \(-0.453631\pi\)
0.145159 + 0.989408i \(0.453631\pi\)
\(108\) 0 0
\(109\) 379.701 0.333658 0.166829 0.985986i \(-0.446647\pi\)
0.166829 + 0.985986i \(0.446647\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1463.88 + 1228.34i −1.21867 + 1.02259i −0.219780 + 0.975549i \(0.570534\pi\)
−0.998893 + 0.0470391i \(0.985021\pi\)
\(114\) 0 0
\(115\) 6.39960 2.32926i 0.00518927 0.00188874i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 402.394 + 337.648i 0.309978 + 0.260102i
\(120\) 0 0
\(121\) 383.283 + 2173.71i 0.287966 + 1.63314i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 690.799 + 1196.50i 0.494295 + 0.856145i
\(126\) 0 0
\(127\) −84.0772 + 145.626i −0.0587453 + 0.101750i −0.893902 0.448262i \(-0.852043\pi\)
0.835157 + 0.550011i \(0.185377\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1884.63 685.949i −1.25695 0.457493i −0.374207 0.927345i \(-0.622085\pi\)
−0.882745 + 0.469852i \(0.844307\pi\)
\(132\) 0 0
\(133\) 85.7348 486.226i 0.0558958 0.317001i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 160.622 910.931i 0.100167 0.568073i −0.892874 0.450306i \(-0.851315\pi\)
0.993041 0.117768i \(-0.0375738\pi\)
\(138\) 0 0
\(139\) 2389.86 + 869.838i 1.45831 + 0.530782i 0.944900 0.327360i \(-0.106159\pi\)
0.513412 + 0.858142i \(0.328381\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −860.732 + 1490.83i −0.503343 + 0.871815i
\(144\) 0 0
\(145\) −301.569 522.332i −0.172717 0.299154i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 353.990 + 2007.58i 0.194631 + 1.10381i 0.912943 + 0.408086i \(0.133804\pi\)
−0.718313 + 0.695720i \(0.755085\pi\)
\(150\) 0 0
\(151\) 631.192 + 529.633i 0.340170 + 0.285437i 0.796829 0.604205i \(-0.206509\pi\)
−0.456658 + 0.889642i \(0.650954\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −78.7217 + 28.6523i −0.0407940 + 0.0148478i
\(156\) 0 0
\(157\) −721.515 + 605.423i −0.366772 + 0.307758i −0.807483 0.589891i \(-0.799171\pi\)
0.440711 + 0.897649i \(0.354726\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.0248 0.00882332
\(162\) 0 0
\(163\) −3395.63 −1.63169 −0.815847 0.578267i \(-0.803729\pi\)
−0.815847 + 0.578267i \(0.803729\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2218.43 1861.48i 1.02795 0.862550i 0.0373420 0.999303i \(-0.488111\pi\)
0.990605 + 0.136752i \(0.0436665\pi\)
\(168\) 0 0
\(169\) 1277.47 464.961i 0.581461 0.211634i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1595.10 + 1338.45i 0.701002 + 0.588211i 0.922059 0.387050i \(-0.126506\pi\)
−0.221056 + 0.975261i \(0.570950\pi\)
\(174\) 0 0
\(175\) 246.355 + 1397.15i 0.106416 + 0.603513i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2179.06 + 3774.24i 0.909890 + 1.57598i 0.814215 + 0.580563i \(0.197168\pi\)
0.0956752 + 0.995413i \(0.469499\pi\)
\(180\) 0 0
\(181\) −1347.22 + 2333.45i −0.553248 + 0.958254i 0.444789 + 0.895635i \(0.353279\pi\)
−0.998038 + 0.0626187i \(0.980055\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2372.27 863.436i −0.942772 0.343141i
\(186\) 0 0
\(187\) 303.053 1718.70i 0.118510 0.672105i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 523.862 2970.97i 0.198457 1.12551i −0.708951 0.705258i \(-0.750831\pi\)
0.907408 0.420250i \(-0.138058\pi\)
\(192\) 0 0
\(193\) 1388.37 + 505.325i 0.517808 + 0.188467i 0.587686 0.809089i \(-0.300039\pi\)
−0.0698784 + 0.997556i \(0.522261\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1889.39 + 3272.53i −0.683318 + 1.18354i 0.290644 + 0.956831i \(0.406131\pi\)
−0.973962 + 0.226711i \(0.927203\pi\)
\(198\) 0 0
\(199\) 160.647 + 278.248i 0.0572258 + 0.0991180i 0.893219 0.449622i \(-0.148441\pi\)
−0.835993 + 0.548740i \(0.815108\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −277.198 1572.07i −0.0958399 0.543535i
\(204\) 0 0
\(205\) −1145.68 961.343i −0.390332 0.327527i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1541.43 + 561.034i −0.510157 + 0.185682i
\(210\) 0 0
\(211\) 973.780 817.098i 0.317714 0.266594i −0.469957 0.882689i \(-0.655731\pi\)
0.787672 + 0.616095i \(0.211286\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3666.15 −1.16293
\(216\) 0 0
\(217\) −221.724 −0.0693622
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 650.448 545.791i 0.197981 0.166126i
\(222\) 0 0
\(223\) 4698.03 1709.94i 1.41078 0.513481i 0.479420 0.877585i \(-0.340847\pi\)
0.931358 + 0.364104i \(0.118625\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 907.775 + 761.713i 0.265423 + 0.222717i 0.765780 0.643103i \(-0.222353\pi\)
−0.500356 + 0.865820i \(0.666798\pi\)
\(228\) 0 0
\(229\) 321.039 + 1820.70i 0.0926412 + 0.525394i 0.995445 + 0.0953410i \(0.0303941\pi\)
−0.902803 + 0.430053i \(0.858495\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1585.36 2745.93i −0.445754 0.772068i 0.552351 0.833612i \(-0.313731\pi\)
−0.998104 + 0.0615437i \(0.980398\pi\)
\(234\) 0 0
\(235\) 2069.73 3584.87i 0.574528 0.995111i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5573.01 2028.41i −1.50832 0.548983i −0.550119 0.835086i \(-0.685418\pi\)
−0.958199 + 0.286104i \(0.907640\pi\)
\(240\) 0 0
\(241\) −517.453 + 2934.62i −0.138307 + 0.784380i 0.834192 + 0.551474i \(0.185935\pi\)
−0.972499 + 0.232906i \(0.925177\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −26.3800 + 149.608i −0.00687900 + 0.0390127i
\(246\) 0 0
\(247\) −749.953 272.961i −0.193192 0.0703161i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3456.30 5986.49i 0.869162 1.50543i 0.00630772 0.999980i \(-0.497992\pi\)
0.862854 0.505453i \(-0.168674\pi\)
\(252\) 0 0
\(253\) −29.9428 51.8624i −0.00744066 0.0128876i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 882.257 + 5003.53i 0.214139 + 1.21444i 0.882395 + 0.470510i \(0.155930\pi\)
−0.668256 + 0.743931i \(0.732959\pi\)
\(258\) 0 0
\(259\) −5118.43 4294.87i −1.22797 1.03039i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7196.97 2619.48i 1.68739 0.614160i 0.693099 0.720843i \(-0.256245\pi\)
0.994293 + 0.106682i \(0.0340228\pi\)
\(264\) 0 0
\(265\) 264.928 222.301i 0.0614129 0.0515316i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6884.46 1.56042 0.780210 0.625518i \(-0.215112\pi\)
0.780210 + 0.625518i \(0.215112\pi\)
\(270\) 0 0
\(271\) −3232.55 −0.724588 −0.362294 0.932064i \(-0.618006\pi\)
−0.362294 + 0.932064i \(0.618006\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3610.75 3029.78i 0.791768 0.664372i
\(276\) 0 0
\(277\) −2971.80 + 1081.65i −0.644615 + 0.234621i −0.643580 0.765379i \(-0.722552\pi\)
−0.00103476 + 0.999999i \(0.500329\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3251.44 2728.28i −0.690266 0.579202i 0.228720 0.973492i \(-0.426546\pi\)
−0.918986 + 0.394290i \(0.870991\pi\)
\(282\) 0 0
\(283\) −927.955 5262.69i −0.194916 1.10542i −0.912538 0.408991i \(-0.865881\pi\)
0.717623 0.696432i \(-0.245230\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1979.18 3428.04i −0.407063 0.705054i
\(288\) 0 0
\(289\) 2026.09 3509.30i 0.412394 0.714288i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2133.39 776.490i −0.425372 0.154823i 0.120459 0.992718i \(-0.461563\pi\)
−0.545830 + 0.837896i \(0.683786\pi\)
\(294\) 0 0
\(295\) 501.229 2842.61i 0.0989243 0.561028i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.05944 28.6935i 0.000978579 0.00554980i
\(300\) 0 0
\(301\) −9118.00 3318.68i −1.74602 0.635500i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1759.12 3046.88i 0.330252 0.572013i
\(306\) 0 0
\(307\) −324.386 561.853i −0.0603052 0.104452i 0.834297 0.551316i \(-0.185874\pi\)
−0.894602 + 0.446864i \(0.852541\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 990.109 + 5615.19i 0.180527 + 1.02382i 0.931569 + 0.363565i \(0.118441\pi\)
−0.751042 + 0.660255i \(0.770448\pi\)
\(312\) 0 0
\(313\) 3737.23 + 3135.91i 0.674890 + 0.566300i 0.914508 0.404567i \(-0.132578\pi\)
−0.239618 + 0.970867i \(0.577022\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6695.08 + 2436.81i −1.18623 + 0.431751i −0.858397 0.512987i \(-0.828539\pi\)
−0.327829 + 0.944737i \(0.606317\pi\)
\(318\) 0 0
\(319\) −4062.80 + 3409.09i −0.713081 + 0.598346i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 809.093 0.139378
\(324\) 0 0
\(325\) 2293.26 0.391407
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8392.68 7042.30i 1.40639 1.18010i
\(330\) 0 0
\(331\) −4873.32 + 1773.74i −0.809250 + 0.294543i −0.713314 0.700844i \(-0.752807\pi\)
−0.0959363 + 0.995387i \(0.530584\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1134.38 + 951.855i 0.185008 + 0.155240i
\(336\) 0 0
\(337\) −1266.75 7184.10i −0.204760 1.16125i −0.897816 0.440370i \(-0.854847\pi\)
0.693056 0.720884i \(-0.256264\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 368.327 + 637.961i 0.0584927 + 0.101312i
\(342\) 0 0
\(343\) −3271.52 + 5666.44i −0.515002 + 0.892009i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3010.74 + 1095.82i 0.465777 + 0.169529i 0.564239 0.825612i \(-0.309170\pi\)
−0.0984612 + 0.995141i \(0.531392\pi\)
\(348\) 0 0
\(349\) −322.003 + 1826.17i −0.0493881 + 0.280094i −0.999493 0.0318364i \(-0.989864\pi\)
0.950105 + 0.311930i \(0.100976\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −57.4469 + 325.798i −0.00866173 + 0.0491231i −0.988832 0.149033i \(-0.952384\pi\)
0.980171 + 0.198156i \(0.0634952\pi\)
\(354\) 0 0
\(355\) 1271.12 + 462.651i 0.190040 + 0.0691689i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2668.13 + 4621.33i −0.392251 + 0.679399i −0.992746 0.120229i \(-0.961637\pi\)
0.600495 + 0.799629i \(0.294970\pi\)
\(360\) 0 0
\(361\) 3049.26 + 5281.47i 0.444563 + 0.770006i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 936.693 + 5312.25i 0.134325 + 0.761797i
\(366\) 0 0
\(367\) 1992.63 + 1672.02i 0.283418 + 0.237816i 0.773403 0.633915i \(-0.218553\pi\)
−0.489984 + 0.871731i \(0.662998\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 860.130 313.062i 0.120366 0.0438096i
\(372\) 0 0
\(373\) 4976.72 4175.97i 0.690844 0.579687i −0.228308 0.973589i \(-0.573319\pi\)
0.919153 + 0.393902i \(0.128875\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2580.37 −0.352508
\(378\) 0 0
\(379\) 5410.32 0.733271 0.366635 0.930365i \(-0.380510\pi\)
0.366635 + 0.930365i \(0.380510\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6463.74 5423.72i 0.862354 0.723601i −0.100120 0.994975i \(-0.531923\pi\)
0.962474 + 0.271374i \(0.0874781\pi\)
\(384\) 0 0
\(385\) −6769.57 + 2463.92i −0.896128 + 0.326164i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8970.57 7527.21i −1.16922 0.981091i −0.169229 0.985577i \(-0.554128\pi\)
−0.999990 + 0.00448556i \(0.998572\pi\)
\(390\) 0 0
\(391\) 5.12924 + 29.0894i 0.000663419 + 0.00376244i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4424.78 7663.95i −0.563633 0.976241i
\(396\) 0 0
\(397\) 6759.21 11707.3i 0.854496 1.48003i −0.0226159 0.999744i \(-0.507199\pi\)
0.877112 0.480286i \(-0.159467\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6064.01 2207.12i −0.755168 0.274859i −0.0643893 0.997925i \(-0.520510\pi\)
−0.690779 + 0.723066i \(0.742732\pi\)
\(402\) 0 0
\(403\) −62.2364 + 352.960i −0.00769284 + 0.0436283i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3854.82 + 21861.7i −0.469475 + 2.66252i
\(408\) 0 0
\(409\) 5073.67 + 1846.66i 0.613390 + 0.223256i 0.629986 0.776606i \(-0.283061\pi\)
−0.0165958 + 0.999862i \(0.505283\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3819.79 6616.07i 0.455108 0.788270i
\(414\) 0 0
\(415\) −217.546 376.800i −0.0257323 0.0445696i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1598.37 + 9064.78i 0.186361 + 1.05691i 0.924194 + 0.381923i \(0.124738\pi\)
−0.737833 + 0.674983i \(0.764151\pi\)
\(420\) 0 0
\(421\) 6648.55 + 5578.80i 0.769669 + 0.645829i 0.940624 0.339450i \(-0.110241\pi\)
−0.170955 + 0.985279i \(0.554685\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2184.69 + 795.162i −0.249348 + 0.0907553i
\(426\) 0 0
\(427\) 7133.18 5985.45i 0.808428 0.678351i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13477.8 −1.50628 −0.753138 0.657862i \(-0.771461\pi\)
−0.753138 + 0.657862i \(0.771461\pi\)
\(432\) 0 0
\(433\) 12659.9 1.40507 0.702533 0.711651i \(-0.252052\pi\)
0.702533 + 0.711651i \(0.252052\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21.2680 17.8460i 0.00232811 0.00195352i
\(438\) 0 0
\(439\) −8751.79 + 3185.39i −0.951480 + 0.346311i −0.770689 0.637211i \(-0.780088\pi\)
−0.180791 + 0.983522i \(0.557866\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3512.25 + 2947.12i 0.376686 + 0.316077i 0.811400 0.584492i \(-0.198706\pi\)
−0.434714 + 0.900569i \(0.643150\pi\)
\(444\) 0 0
\(445\) 1095.58 + 6213.32i 0.116709 + 0.661887i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −530.524 918.895i −0.0557616 0.0965820i 0.836797 0.547513i \(-0.184425\pi\)
−0.892559 + 0.450931i \(0.851092\pi\)
\(450\) 0 0
\(451\) −6575.61 + 11389.3i −0.686548 + 1.18914i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3293.61 1198.77i −0.339355 0.123515i
\(456\) 0 0
\(457\) −2136.23 + 12115.2i −0.218662 + 1.24009i 0.655775 + 0.754956i \(0.272342\pi\)
−0.874437 + 0.485139i \(0.838769\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.2863 171.762i 0.00305981 0.0173531i −0.983240 0.182318i \(-0.941640\pi\)
0.986299 + 0.164965i \(0.0527511\pi\)
\(462\) 0 0
\(463\) 8366.64 + 3045.21i 0.839808 + 0.305665i 0.725878 0.687824i \(-0.241434\pi\)
0.113930 + 0.993489i \(0.463656\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1326.26 + 2297.15i −0.131418 + 0.227622i −0.924223 0.381853i \(-0.875286\pi\)
0.792806 + 0.609475i \(0.208620\pi\)
\(468\) 0 0
\(469\) 1959.64 + 3394.20i 0.192938 + 0.334179i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5598.02 + 31748.0i 0.544181 + 3.08620i
\(474\) 0 0
\(475\) 1673.97 + 1404.63i 0.161699 + 0.135681i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18164.9 + 6611.48i −1.73272 + 0.630660i −0.998819 0.0485928i \(-0.984526\pi\)
−0.733905 + 0.679252i \(0.762304\pi\)
\(480\) 0 0
\(481\) −8273.67 + 6942.43i −0.784297 + 0.658103i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8084.29 0.756884
\(486\) 0 0
\(487\) 6386.49 0.594249 0.297125 0.954839i \(-0.403972\pi\)
0.297125 + 0.954839i \(0.403972\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13082.9 10977.9i 1.20249 1.00901i 0.202936 0.979192i \(-0.434952\pi\)
0.999555 0.0298176i \(-0.00949265\pi\)
\(492\) 0 0
\(493\) 2458.20 894.712i 0.224568 0.0817359i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2742.58 + 2301.30i 0.247528 + 0.207701i
\(498\) 0 0
\(499\) −1145.21 6494.83i −0.102739 0.582662i −0.992099 0.125455i \(-0.959961\pi\)
0.889360 0.457207i \(-0.151150\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3417.70 + 5919.63i 0.302958 + 0.524738i 0.976805 0.214133i \(-0.0686925\pi\)
−0.673847 + 0.738871i \(0.735359\pi\)
\(504\) 0 0
\(505\) 3362.17 5823.45i 0.296267 0.513149i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 435.292 + 158.433i 0.0379056 + 0.0137965i 0.360903 0.932603i \(-0.382468\pi\)
−0.322998 + 0.946400i \(0.604691\pi\)
\(510\) 0 0
\(511\) −2479.14 + 14059.9i −0.214620 + 1.21717i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −386.650 + 2192.80i −0.0330831 + 0.187624i
\(516\) 0 0
\(517\) −34204.5 12449.4i −2.90970 1.05904i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5085.58 8808.49i 0.427646 0.740704i −0.569018 0.822325i \(-0.692676\pi\)
0.996663 + 0.0816212i \(0.0260098\pi\)
\(522\) 0 0
\(523\) 4957.61 + 8586.84i 0.414496 + 0.717928i 0.995375 0.0960616i \(-0.0306246\pi\)
−0.580879 + 0.813990i \(0.697291\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −63.0949 357.829i −0.00521529 0.0295774i
\(528\) 0 0
\(529\) −9319.69 7820.15i −0.765981 0.642734i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6012.60 + 2188.41i −0.488620 + 0.177843i
\(534\) 0 0
\(535\) 1665.11 1397.19i 0.134559 0.112908i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1335.85 0.106752
\(540\) 0 0
\(541\) 12272.1 0.975264 0.487632 0.873049i \(-0.337861\pi\)
0.487632 + 0.873049i \(0.337861\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1967.59 1651.00i 0.154646 0.129764i
\(546\) 0 0
\(547\) 17044.4 6203.64i 1.33229 0.484915i 0.424915 0.905233i \(-0.360304\pi\)
0.907377 + 0.420318i \(0.138082\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1883.54 1580.48i −0.145629 0.122197i
\(552\) 0 0
\(553\) −4067.21 23066.3i −0.312758 1.77374i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1502.66 2602.69i −0.114309 0.197988i 0.803195 0.595717i \(-0.203132\pi\)
−0.917503 + 0.397729i \(0.869799\pi\)
\(558\) 0 0
\(559\) −7842.33 + 13583.3i −0.593373 + 1.02775i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2205.13 802.602i −0.165071 0.0600811i 0.258163 0.966101i \(-0.416883\pi\)
−0.423234 + 0.906020i \(0.639105\pi\)
\(564\) 0 0
\(565\) −2244.71 + 12730.4i −0.167143 + 0.947913i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3585.08 20332.0i 0.264138 1.49800i −0.507341 0.861745i \(-0.669372\pi\)
0.771479 0.636255i \(-0.219517\pi\)
\(570\) 0 0
\(571\) 10022.8 + 3648.01i 0.734575 + 0.267363i 0.682100 0.731259i \(-0.261067\pi\)
0.0524748 + 0.998622i \(0.483289\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −39.8885 + 69.0889i −0.00289298 + 0.00501079i
\(576\) 0 0
\(577\) −5276.40 9138.99i −0.380692 0.659378i 0.610469 0.792040i \(-0.290981\pi\)
−0.991161 + 0.132662i \(0.957648\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −199.965 1134.06i −0.0142787 0.0809788i
\(582\) 0 0
\(583\) −2329.61 1954.78i −0.165493 0.138865i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5519.40 2008.90i 0.388092 0.141254i −0.140602 0.990066i \(-0.544904\pi\)
0.528695 + 0.848812i \(0.322682\pi\)
\(588\) 0 0
\(589\) −261.618 + 219.524i −0.0183018 + 0.0153571i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10619.0 0.735360 0.367680 0.929952i \(-0.380152\pi\)
0.367680 + 0.929952i \(0.380152\pi\)
\(594\) 0 0
\(595\) 3553.33 0.244828
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 291.282 244.414i 0.0198689 0.0166719i −0.632799 0.774316i \(-0.718094\pi\)
0.652668 + 0.757644i \(0.273650\pi\)
\(600\) 0 0
\(601\) −2760.03 + 1004.57i −0.187327 + 0.0681816i −0.433981 0.900922i \(-0.642891\pi\)
0.246653 + 0.969104i \(0.420669\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11437.8 + 9597.45i 0.768616 + 0.644945i
\(606\) 0 0
\(607\) 4004.42 + 22710.2i 0.267766 + 1.51858i 0.761040 + 0.648705i \(0.224689\pi\)
−0.493274 + 0.869874i \(0.664200\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8854.79 15336.9i −0.586295 1.01549i
\(612\) 0 0
\(613\) 7919.71 13717.3i 0.521818 0.903814i −0.477860 0.878436i \(-0.658588\pi\)
0.999678 0.0253787i \(-0.00807915\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7369.88 2682.42i −0.480875 0.175024i 0.0901973 0.995924i \(-0.471250\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(618\) 0 0
\(619\) 3859.29 21887.1i 0.250594 1.42119i −0.556539 0.830821i \(-0.687871\pi\)
0.807133 0.590369i \(-0.201018\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2899.66 + 16444.8i −0.186472 + 1.05754i
\(624\) 0 0
\(625\) −525.504 191.268i −0.0336323 0.0122411i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5474.75 9482.54i 0.347047 0.601103i
\(630\) 0 0
\(631\) −2964.46 5134.59i −0.187026 0.323938i 0.757232 0.653146i \(-0.226551\pi\)
−0.944257 + 0.329209i \(0.893218\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 197.523 + 1120.21i 0.0123440 + 0.0700065i
\(636\) 0 0
\(637\) 497.879 + 417.770i 0.0309681 + 0.0259853i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3853.21 + 1402.45i −0.237430 + 0.0864175i −0.457995 0.888955i \(-0.651432\pi\)
0.220565 + 0.975372i \(0.429210\pi\)
\(642\) 0 0
\(643\) 237.689 199.444i 0.0145778 0.0122322i −0.635470 0.772126i \(-0.719193\pi\)
0.650047 + 0.759894i \(0.274749\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17190.2 −1.04454 −0.522268 0.852781i \(-0.674914\pi\)
−0.522268 + 0.852781i \(0.674914\pi\)
\(648\) 0 0
\(649\) −25381.7 −1.53516
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5085.19 4266.98i 0.304746 0.255712i −0.477571 0.878593i \(-0.658482\pi\)
0.782316 + 0.622881i \(0.214038\pi\)
\(654\) 0 0
\(655\) −12748.7 + 4640.13i −0.760506 + 0.276802i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1245.58 + 1045.16i 0.0736279 + 0.0617812i 0.678859 0.734269i \(-0.262475\pi\)
−0.605231 + 0.796050i \(0.706919\pi\)
\(660\) 0 0
\(661\) −1371.31 7777.07i −0.0806924 0.457629i −0.998203 0.0599185i \(-0.980916\pi\)
0.917511 0.397711i \(-0.130195\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1669.92 2892.39i −0.0973785 0.168665i
\(666\) 0 0
\(667\) 44.8823 77.7385i 0.00260547 0.00451281i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −29071.4 10581.1i −1.67256 0.608763i
\(672\) 0 0
\(673\) 2739.40 15535.9i 0.156904 0.889846i −0.800121 0.599839i \(-0.795231\pi\)
0.957025 0.290007i \(-0.0936576\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5133.84 29115.5i 0.291447 1.65288i −0.389856 0.920876i \(-0.627475\pi\)
0.681303 0.732002i \(-0.261414\pi\)
\(678\) 0 0
\(679\) 20106.3 + 7318.09i 1.13639 + 0.413612i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12777.3 + 22131.0i −0.715829 + 1.23985i 0.246810 + 0.969064i \(0.420618\pi\)
−0.962639 + 0.270788i \(0.912716\pi\)
\(684\) 0 0
\(685\) −3128.55 5418.80i −0.174505 0.302251i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −256.927 1457.11i −0.0142063 0.0805680i
\(690\) 0 0
\(691\) −16424.3 13781.6i −0.904212 0.758724i 0.0667969 0.997767i \(-0.478722\pi\)
−0.971009 + 0.239042i \(0.923166\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16166.3 5884.06i 0.882336 0.321144i
\(696\) 0 0
\(697\) 4969.13 4169.60i 0.270042 0.226592i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11587.1 0.624308 0.312154 0.950031i \(-0.398949\pi\)
0.312154 + 0.950031i \(0.398949\pi\)
\(702\) 0 0
\(703\) −10291.6 −0.552142
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13633.5 11439.9i 0.725235 0.608544i
\(708\) 0 0
\(709\) 8347.62 3038.29i 0.442174 0.160938i −0.111332 0.993783i \(-0.535512\pi\)
0.553506 + 0.832845i \(0.313289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.55107 8.01430i −0.000501669 0.000420951i
\(714\) 0 0
\(715\) 2022.12 + 11468.0i 0.105766 + 0.599831i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4391.89 + 7606.97i 0.227802 + 0.394565i 0.957156 0.289571i \(-0.0935127\pi\)
−0.729354 + 0.684136i \(0.760179\pi\)
\(720\) 0 0
\(721\) −2946.60 + 5103.66i −0.152201 + 0.263620i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6639.15 + 2416.45i 0.340099 + 0.123786i
\(726\) 0 0
\(727\) 2189.91 12419.6i 0.111718 0.633587i −0.876604 0.481212i \(-0.840197\pi\)
0.988323 0.152375i \(-0.0486921\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2761.19 15659.5i 0.139707 0.792320i
\(732\) 0 0
\(733\) 5389.52 + 1961.62i 0.271578 + 0.0988462i 0.474219 0.880407i \(-0.342730\pi\)
−0.202642 + 0.979253i \(0.564953\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6510.71 11276.9i 0.325407 0.563622i
\(738\) 0 0
\(739\) −6949.01 12036.0i −0.345904 0.599124i 0.639613 0.768697i \(-0.279095\pi\)
−0.985518 + 0.169573i \(0.945761\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2458.80 13944.6i −0.121406 0.688528i −0.983378 0.181572i \(-0.941882\pi\)
0.861972 0.506956i \(-0.169230\pi\)
\(744\) 0 0
\(745\) 10563.6 + 8863.95i 0.519492 + 0.435906i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5406.04 1967.64i 0.263728 0.0959892i
\(750\) 0 0
\(751\) −8216.97 + 6894.86i −0.399256 + 0.335016i −0.820206 0.572068i \(-0.806141\pi\)
0.420950 + 0.907084i \(0.361697\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5573.74 0.268674
\(756\) 0 0
\(757\) −12941.7 −0.621366 −0.310683 0.950514i \(-0.600558\pi\)
−0.310683 + 0.950514i \(0.600558\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16885.2 14168.4i 0.804323 0.674907i −0.144923 0.989443i \(-0.546293\pi\)
0.949245 + 0.314536i \(0.101849\pi\)
\(762\) 0 0
\(763\) 6388.07 2325.07i 0.303098 0.110319i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9459.87 7937.77i −0.445340 0.373685i
\(768\) 0 0
\(769\) −2620.50 14861.6i −0.122884 0.696908i −0.982542 0.186039i \(-0.940435\pi\)
0.859659 0.510869i \(-0.170676\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16889.2 + 29253.0i 0.785852 + 1.36113i 0.928489 + 0.371360i \(0.121108\pi\)
−0.142637 + 0.989775i \(0.545558\pi\)
\(774\) 0 0
\(775\) 490.669 849.864i 0.0227424 0.0393910i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5729.31 2085.30i −0.263509 0.0959095i
\(780\) 0 0
\(781\) 2065.51 11714.1i 0.0946347 0.536700i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1106.37 + 6274.53i −0.0503032 + 0.285284i
\(786\) 0 0
\(787\) −25567.2 9305.71i −1.15803 0.421490i −0.309640 0.950854i \(-0.600208\pi\)
−0.848395 + 0.529364i \(0.822431\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −17106.6 + 29629.5i −0.768952 + 1.33186i
\(792\) 0 0
\(793\) −7525.94 13035.3i −0.337016 0.583729i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2911.06 16509.5i −0.129379 0.733746i −0.978610 0.205724i \(-0.934045\pi\)
0.849231 0.528022i \(-0.177066\pi\)
\(798\) 0 0
\(799\) 13753.5 + 11540.5i 0.608965 + 0.510982i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 44572.6 16223.1i 1.95882 0.712952i
\(804\) 0 0
\(805\) 93.4036 78.3749i 0.00408950 0.00343150i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8398.15 −0.364973 −0.182487 0.983208i \(-0.558415\pi\)
−0.182487 + 0.983208i \(0.558415\pi\)
\(810\) 0 0
\(811\) −38336.5 −1.65990 −0.829949 0.557839i \(-0.811631\pi\)
−0.829949 + 0.557839i \(0.811631\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −17596.0 + 14764.8i −0.756269 + 0.634585i
\(816\) 0 0
\(817\) −14044.3 + 5111.72i −0.601406 + 0.218894i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21752.9 18252.8i −0.924703 0.775918i 0.0501561 0.998741i \(-0.484028\pi\)
−0.974859 + 0.222824i \(0.928473\pi\)
\(822\) 0 0
\(823\) 4570.67 + 25921.6i 0.193589 + 1.09790i 0.914414 + 0.404780i \(0.132652\pi\)
−0.720825 + 0.693117i \(0.756237\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8653.68 + 14988.6i 0.363867 + 0.630236i 0.988594 0.150608i \(-0.0481230\pi\)
−0.624727 + 0.780843i \(0.714790\pi\)
\(828\) 0 0
\(829\) −3848.42 + 6665.66i −0.161232 + 0.279262i −0.935311 0.353827i \(-0.884880\pi\)
0.774079 + 0.633089i \(0.218213\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −619.164 225.357i −0.0257536 0.00937354i
\(834\) 0 0
\(835\) 3401.74 19292.2i 0.140984 0.799562i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2024.34 + 11480.6i −0.0832992 + 0.472413i 0.914411 + 0.404786i \(0.132654\pi\)
−0.997711 + 0.0676272i \(0.978457\pi\)
\(840\) 0 0
\(841\) 15447.8 + 5622.55i 0.633393 + 0.230536i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4598.05 7964.05i 0.187192 0.324227i
\(846\) 0 0
\(847\) 19758.9 + 34223.4i 0.801562 + 1.38835i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −65.2437 370.015i −0.00262811 0.0149048i
\(852\) 0 0
\(853\) 16135.3 + 13539.1i 0.647668 + 0.543458i 0.906362 0.422501i \(-0.138848\pi\)
−0.258694 + 0.965959i \(0.583292\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10298.8 + 3748.45i −0.410501 + 0.149410i −0.539012 0.842298i \(-0.681202\pi\)
0.128511 + 0.991708i \(0.458980\pi\)
\(858\) 0 0
\(859\) −5778.78 + 4848.97i −0.229534 + 0.192602i −0.750300 0.661098i \(-0.770091\pi\)
0.520766 + 0.853699i \(0.325646\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26268.2 1.03613 0.518065 0.855341i \(-0.326652\pi\)
0.518065 + 0.855341i \(0.326652\pi\)
\(864\) 0 0
\(865\) 14085.5 0.553668
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −59611.6 + 50020.1i −2.32703 + 1.95261i
\(870\) 0 0
\(871\) 5953.26 2166.81i 0.231594 0.0842934i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18948.6 + 15899.8i 0.732093 + 0.614299i
\(876\) 0 0
\(877\) 8371.42 + 47476.7i 0.322329 + 1.82802i 0.527814 + 0.849360i \(0.323012\pi\)
−0.205485 + 0.978660i \(0.565877\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8868.20 + 15360.2i 0.339134 + 0.587398i 0.984270 0.176670i \(-0.0565326\pi\)
−0.645136 + 0.764068i \(0.723199\pi\)
\(882\) 0 0
\(883\) 17748.2 30740.7i 0.676413 1.17158i −0.299640 0.954052i \(-0.596867\pi\)
0.976054 0.217530i \(-0.0698000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2797.75 + 1018.30i 0.105907 + 0.0385469i 0.394430 0.918926i \(-0.370942\pi\)
−0.288523 + 0.957473i \(0.593164\pi\)
\(888\) 0 0
\(889\) −522.783 + 2964.85i −0.0197228 + 0.111854i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2930.34 16618.8i 0.109810 0.622762i
\(894\) 0 0
\(895\) 27702.8 + 10083.0i 1.03464 + 0.376578i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −552.099 + 956.264i −0.0204822 + 0.0354763i
\(900\) 0 0
\(901\) 749.998 + 1299.03i 0.0277315 + 0.0480323i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3165.02 + 17949.7i 0.116253 + 0.659303i
\(906\) 0 0
\(907\) −3444.78 2890.51i −0.126110 0.105819i 0.577551 0.816354i \(-0.304008\pi\)
−0.703661 + 0.710535i \(0.748453\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3923.42 1428.01i 0.142688 0.0519342i −0.269689 0.962947i \(-0.586921\pi\)
0.412377 + 0.911013i \(0.364699\pi\)
\(912\) 0 0
\(913\) −2930.82 + 2459.25i −0.106239 + 0.0891449i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −35907.3 −1.29309
\(918\) 0 0
\(919\) −25666.6 −0.921287 −0.460643 0.887585i \(-0.652381\pi\)
−0.460643 + 0.887585i \(0.652381\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4433.24 3719.93i 0.158095 0.132658i
\(924\) 0 0
\(925\) 27789.1 10114.4i 0.987785 0.359524i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 20279.1 + 17016.2i 0.716185 + 0.600950i 0.926327 0.376720i \(-0.122948\pi\)
−0.210142 + 0.977671i \(0.567393\pi\)
\(930\) 0 0
\(931\) 107.542 + 609.903i 0.00378578 + 0.0214702i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5902.78 10223.9i −0.206462 0.357602i
\(936\) 0 0
\(937\) −17460.8 + 30243.1i −0.608774 + 1.05443i 0.382669 + 0.923885i \(0.375005\pi\)
−0.991443 + 0.130542i \(0.958328\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −46488.1 16920.3i −1.61049 0.586169i −0.628950 0.777446i \(-0.716515\pi\)
−0.981536 + 0.191276i \(0.938737\pi\)
\(942\) 0 0
\(943\) 38.6519 219.206i 0.00133476 0.00756980i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4727.28 + 26809.7i −0.162213 + 0.919956i 0.789678 + 0.613521i \(0.210247\pi\)
−0.951891 + 0.306435i \(0.900864\pi\)
\(948\) 0 0
\(949\) 21685.9 + 7893.04i 0.741787 + 0.269988i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −15252.7 + 26418.4i −0.518450 + 0.897982i 0.481320 + 0.876545i \(0.340158\pi\)
−0.999770 + 0.0214371i \(0.993176\pi\)
\(954\) 0 0
\(955\) −10203.7 17673.3i −0.345741 0.598841i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2875.72 16309.0i −0.0968320 0.549161i
\(960\) 0 0
\(961\) −22703.7 19050.7i −0.762101 0.639478i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9391.68 3418.29i 0.313294 0.114030i
\(966\) 0 0
\(967\) 11428.8 9589.87i 0.380066 0.318914i −0.432662 0.901556i \(-0.642426\pi\)
0.812728 + 0.582643i \(0.197981\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −16017.3 −0.529373 −0.264686 0.964335i \(-0.585268\pi\)
−0.264686 + 0.964335i \(0.585268\pi\)
\(972\) 0 0
\(973\) 45533.3 1.50024
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −21806.5 + 18297.8i −0.714075 + 0.599180i −0.925739 0.378163i \(-0.876556\pi\)
0.211664 + 0.977342i \(0.432112\pi\)
\(978\) 0 0
\(979\) 52133.1 18974.9i 1.70192 0.619448i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −18350.6 15398.0i −0.595417 0.499614i 0.294552 0.955635i \(-0.404830\pi\)
−0.889969 + 0.456022i \(0.849274\pi\)
\(984\) 0 0
\(985\) 4438.76 + 25173.5i 0.143584 + 0.814307i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −272.816 472.530i −0.00877152 0.0151927i
\(990\) 0 0
\(991\) −8682.28 + 15038.1i −0.278306 + 0.482041i −0.970964 0.239226i \(-0.923106\pi\)
0.692658 + 0.721267i \(0.256440\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2042.33 + 743.348i 0.0650716 + 0.0236841i
\(996\) 0 0
\(997\) −3733.89 + 21175.9i −0.118609 + 0.672667i 0.866290 + 0.499541i \(0.166498\pi\)
−0.984900 + 0.173126i \(0.944613\pi\)
\(998\) 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 324.4.i.a.73.6 54
3.2 odd 2 108.4.i.a.25.8 yes 54
27.13 even 9 inner 324.4.i.a.253.6 54
27.14 odd 18 108.4.i.a.13.8 54
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.i.a.13.8 54 27.14 odd 18
108.4.i.a.25.8 yes 54 3.2 odd 2
324.4.i.a.73.6 54 1.1 even 1 trivial
324.4.i.a.253.6 54 27.13 even 9 inner