Properties

Label 144.5.q.a.65.3
Level $144$
Weight $5$
Character 144.65
Analytic conductor $14.885$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,5,Mod(65,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), 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: \(14.8852746841\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.39400128.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.3
Root \(-1.28901 - 2.23263i\) of defining polynomial
Character \(\chi\) \(=\) 144.65
Dual form 144.5.q.a.113.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(8.70335 + 2.29167i) q^{3} +(13.8760 - 8.01130i) q^{5} +(36.2418 - 62.7727i) q^{7} +(70.4965 + 39.8904i) q^{9} +(-83.2749 - 48.0788i) q^{11} +(-76.9530 - 133.286i) q^{13} +(139.127 - 37.9260i) q^{15} -72.7905i q^{17} +190.660 q^{19} +(459.280 - 463.279i) q^{21} +(12.5186 - 7.22762i) q^{23} +(-184.138 + 318.937i) q^{25} +(522.140 + 508.734i) q^{27} +(620.301 + 358.131i) q^{29} +(-151.284 - 262.031i) q^{31} +(-614.590 - 609.285i) q^{33} -1161.38i q^{35} +826.277 q^{37} +(-364.300 - 1336.39i) q^{39} +(481.613 - 278.059i) q^{41} +(446.340 - 773.084i) q^{43} +(1297.78 - 11.2508i) q^{45} +(3425.50 + 1977.72i) q^{47} +(-1426.44 - 2470.67i) q^{49} +(166.812 - 633.521i) q^{51} +1966.96i q^{53} -1540.69 q^{55} +(1659.38 + 436.930i) q^{57} +(-4689.48 + 2707.47i) q^{59} +(856.210 - 1483.00i) q^{61} +(5058.95 - 2979.56i) q^{63} +(-2135.60 - 1232.99i) q^{65} +(-2317.24 - 4013.58i) q^{67} +(125.517 - 34.2160i) q^{69} +6697.12i q^{71} -4823.86 q^{73} +(-2333.51 + 2353.83i) q^{75} +(-6036.07 + 3484.93i) q^{77} +(-2864.40 + 4961.28i) q^{79} +(3378.52 + 5624.26i) q^{81} +(-2452.78 - 1416.12i) q^{83} +(-583.147 - 1010.04i) q^{85} +(4577.98 + 4538.46i) q^{87} +14277.7i q^{89} -11155.7 q^{91} +(-716.187 - 2627.24i) q^{93} +(2645.60 - 1527.44i) q^{95} +(-3582.65 + 6205.34i) q^{97} +(-3952.71 - 6711.25i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 12 q^{5} - 12 q^{7} + 99 q^{9} - 483 q^{11} - 6 q^{13} + 1026 q^{15} + 258 q^{19} + 480 q^{21} + 282 q^{23} - 273 q^{25} - 54 q^{27} - 1056 q^{29} - 1290 q^{31} + 279 q^{33} + 12 q^{37}+ \cdots + 9126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(3\) 8.70335 + 2.29167i 0.967039 + 0.254630i
\(4\) 0 0
\(5\) 13.8760 8.01130i 0.555039 0.320452i −0.196113 0.980581i \(-0.562832\pi\)
0.751152 + 0.660129i \(0.229499\pi\)
\(6\) 0 0
\(7\) 36.2418 62.7727i 0.739630 1.28108i −0.213033 0.977045i \(-0.568334\pi\)
0.952662 0.304031i \(-0.0983326\pi\)
\(8\) 0 0
\(9\) 70.4965 + 39.8904i 0.870327 + 0.492474i
\(10\) 0 0
\(11\) −83.2749 48.0788i −0.688222 0.397345i 0.114723 0.993397i \(-0.463402\pi\)
−0.802946 + 0.596052i \(0.796735\pi\)
\(12\) 0 0
\(13\) −76.9530 133.286i −0.455343 0.788677i 0.543365 0.839497i \(-0.317150\pi\)
−0.998708 + 0.0508193i \(0.983817\pi\)
\(14\) 0 0
\(15\) 139.127 37.9260i 0.618341 0.168560i
\(16\) 0 0
\(17\) 72.7905i 0.251870i −0.992038 0.125935i \(-0.959807\pi\)
0.992038 0.125935i \(-0.0401931\pi\)
\(18\) 0 0
\(19\) 190.660 0.528145 0.264072 0.964503i \(-0.414934\pi\)
0.264072 + 0.964503i \(0.414934\pi\)
\(20\) 0 0
\(21\) 459.280 463.279i 1.04145 1.05052i
\(22\) 0 0
\(23\) 12.5186 7.22762i 0.0236647 0.0136628i −0.488121 0.872776i \(-0.662318\pi\)
0.511786 + 0.859113i \(0.328984\pi\)
\(24\) 0 0
\(25\) −184.138 + 318.937i −0.294621 + 0.510298i
\(26\) 0 0
\(27\) 522.140 + 508.734i 0.716242 + 0.697852i
\(28\) 0 0
\(29\) 620.301 + 358.131i 0.737575 + 0.425839i 0.821187 0.570659i \(-0.193312\pi\)
−0.0836117 + 0.996498i \(0.526646\pi\)
\(30\) 0 0
\(31\) −151.284 262.031i −0.157423 0.272665i 0.776515 0.630098i \(-0.216985\pi\)
−0.933939 + 0.357433i \(0.883652\pi\)
\(32\) 0 0
\(33\) −614.590 609.285i −0.564362 0.559490i
\(34\) 0 0
\(35\) 1161.38i 0.948063i
\(36\) 0 0
\(37\) 826.277 0.603562 0.301781 0.953377i \(-0.402419\pi\)
0.301781 + 0.953377i \(0.402419\pi\)
\(38\) 0 0
\(39\) −364.300 1336.39i −0.239514 0.878625i
\(40\) 0 0
\(41\) 481.613 278.059i 0.286504 0.165413i −0.349860 0.936802i \(-0.613771\pi\)
0.636364 + 0.771389i \(0.280438\pi\)
\(42\) 0 0
\(43\) 446.340 773.084i 0.241395 0.418109i −0.719717 0.694268i \(-0.755728\pi\)
0.961112 + 0.276159i \(0.0890616\pi\)
\(44\) 0 0
\(45\) 1297.78 11.2508i 0.640880 0.00555597i
\(46\) 0 0
\(47\) 3425.50 + 1977.72i 1.55070 + 0.895299i 0.998085 + 0.0618654i \(0.0197050\pi\)
0.552619 + 0.833434i \(0.313628\pi\)
\(48\) 0 0
\(49\) −1426.44 2470.67i −0.594104 1.02902i
\(50\) 0 0
\(51\) 166.812 633.521i 0.0641337 0.243568i
\(52\) 0 0
\(53\) 1966.96i 0.700234i 0.936706 + 0.350117i \(0.113858\pi\)
−0.936706 + 0.350117i \(0.886142\pi\)
\(54\) 0 0
\(55\) −1540.69 −0.509320
\(56\) 0 0
\(57\) 1659.38 + 436.930i 0.510736 + 0.134481i
\(58\) 0 0
\(59\) −4689.48 + 2707.47i −1.34716 + 0.777785i −0.987847 0.155431i \(-0.950323\pi\)
−0.359316 + 0.933216i \(0.616990\pi\)
\(60\) 0 0
\(61\) 856.210 1483.00i 0.230102 0.398549i −0.727736 0.685858i \(-0.759427\pi\)
0.957838 + 0.287309i \(0.0927606\pi\)
\(62\) 0 0
\(63\) 5058.95 2979.56i 1.27462 0.750707i
\(64\) 0 0
\(65\) −2135.60 1232.99i −0.505467 0.291831i
\(66\) 0 0
\(67\) −2317.24 4013.58i −0.516205 0.894093i −0.999823 0.0188141i \(-0.994011\pi\)
0.483618 0.875279i \(-0.339322\pi\)
\(68\) 0 0
\(69\) 125.517 34.2160i 0.0263636 0.00718672i
\(70\) 0 0
\(71\) 6697.12i 1.32853i 0.747497 + 0.664265i \(0.231256\pi\)
−0.747497 + 0.664265i \(0.768744\pi\)
\(72\) 0 0
\(73\) −4823.86 −0.905208 −0.452604 0.891712i \(-0.649505\pi\)
−0.452604 + 0.891712i \(0.649505\pi\)
\(74\) 0 0
\(75\) −2333.51 + 2353.83i −0.414847 + 0.418459i
\(76\) 0 0
\(77\) −6036.07 + 3484.93i −1.01806 + 0.587777i
\(78\) 0 0
\(79\) −2864.40 + 4961.28i −0.458964 + 0.794950i −0.998907 0.0467524i \(-0.985113\pi\)
0.539942 + 0.841702i \(0.318446\pi\)
\(80\) 0 0
\(81\) 3378.52 + 5624.26i 0.514939 + 0.857227i
\(82\) 0 0
\(83\) −2452.78 1416.12i −0.356044 0.205562i 0.311300 0.950312i \(-0.399235\pi\)
−0.667344 + 0.744750i \(0.732569\pi\)
\(84\) 0 0
\(85\) −583.147 1010.04i −0.0807124 0.139798i
\(86\) 0 0
\(87\) 4577.98 + 4538.46i 0.604832 + 0.599612i
\(88\) 0 0
\(89\) 14277.7i 1.80251i 0.433290 + 0.901255i \(0.357353\pi\)
−0.433290 + 0.901255i \(0.642647\pi\)
\(90\) 0 0
\(91\) −11155.7 −1.34714
\(92\) 0 0
\(93\) −716.187 2627.24i −0.0828058 0.303763i
\(94\) 0 0
\(95\) 2645.60 1527.44i 0.293141 0.169245i
\(96\) 0 0
\(97\) −3582.65 + 6205.34i −0.380769 + 0.659511i −0.991172 0.132580i \(-0.957674\pi\)
0.610403 + 0.792091i \(0.291007\pi\)
\(98\) 0 0
\(99\) −3952.71 6711.25i −0.403297 0.684752i
\(100\) 0 0
\(101\) −1696.72 979.600i −0.166328 0.0960298i 0.414525 0.910038i \(-0.363948\pi\)
−0.580853 + 0.814008i \(0.697281\pi\)
\(102\) 0 0
\(103\) −2577.74 4464.77i −0.242976 0.420848i 0.718584 0.695440i \(-0.244790\pi\)
−0.961561 + 0.274592i \(0.911457\pi\)
\(104\) 0 0
\(105\) 2661.49 10107.9i 0.241405 0.916814i
\(106\) 0 0
\(107\) 9117.08i 0.796321i 0.917316 + 0.398161i \(0.130351\pi\)
−0.917316 + 0.398161i \(0.869649\pi\)
\(108\) 0 0
\(109\) −16161.1 −1.36024 −0.680122 0.733099i \(-0.738073\pi\)
−0.680122 + 0.733099i \(0.738073\pi\)
\(110\) 0 0
\(111\) 7191.38 + 1893.55i 0.583668 + 0.153685i
\(112\) 0 0
\(113\) 18272.5 10549.6i 1.43100 0.826189i 0.433804 0.901007i \(-0.357171\pi\)
0.997197 + 0.0748185i \(0.0238377\pi\)
\(114\) 0 0
\(115\) 115.805 200.581i 0.00875654 0.0151668i
\(116\) 0 0
\(117\) −108.071 12465.9i −0.00789470 0.910652i
\(118\) 0 0
\(119\) −4569.26 2638.06i −0.322665 0.186291i
\(120\) 0 0
\(121\) −2697.36 4671.97i −0.184233 0.319102i
\(122\) 0 0
\(123\) 4828.87 1316.35i 0.319180 0.0870084i
\(124\) 0 0
\(125\) 15914.9i 1.01855i
\(126\) 0 0
\(127\) 20660.9 1.28098 0.640489 0.767968i \(-0.278732\pi\)
0.640489 + 0.767968i \(0.278732\pi\)
\(128\) 0 0
\(129\) 5656.31 5705.55i 0.339902 0.342861i
\(130\) 0 0
\(131\) −2772.53 + 1600.72i −0.161560 + 0.0932768i −0.578600 0.815611i \(-0.696401\pi\)
0.417040 + 0.908888i \(0.363067\pi\)
\(132\) 0 0
\(133\) 6909.88 11968.3i 0.390631 0.676594i
\(134\) 0 0
\(135\) 11320.8 + 2876.17i 0.621170 + 0.157814i
\(136\) 0 0
\(137\) −3353.18 1935.96i −0.178655 0.103147i 0.408005 0.912980i \(-0.366224\pi\)
−0.586661 + 0.809833i \(0.699558\pi\)
\(138\) 0 0
\(139\) −5839.62 10114.5i −0.302242 0.523499i 0.674401 0.738365i \(-0.264402\pi\)
−0.976644 + 0.214866i \(0.931068\pi\)
\(140\) 0 0
\(141\) 25281.1 + 25062.9i 1.27162 + 1.26064i
\(142\) 0 0
\(143\) 14799.2i 0.723714i
\(144\) 0 0
\(145\) 11476.4 0.545844
\(146\) 0 0
\(147\) −6752.87 24772.1i −0.312503 1.14638i
\(148\) 0 0
\(149\) −13069.8 + 7545.88i −0.588705 + 0.339889i −0.764585 0.644522i \(-0.777056\pi\)
0.175880 + 0.984412i \(0.443723\pi\)
\(150\) 0 0
\(151\) 15127.7 26201.9i 0.663465 1.14915i −0.316234 0.948681i \(-0.602419\pi\)
0.979699 0.200474i \(-0.0642481\pi\)
\(152\) 0 0
\(153\) 2903.64 5131.48i 0.124040 0.219210i
\(154\) 0 0
\(155\) −4198.42 2423.96i −0.174752 0.100893i
\(156\) 0 0
\(157\) −10311.4 17859.8i −0.418328 0.724565i 0.577443 0.816431i \(-0.304050\pi\)
−0.995771 + 0.0918653i \(0.970717\pi\)
\(158\) 0 0
\(159\) −4507.61 + 17119.1i −0.178300 + 0.677153i
\(160\) 0 0
\(161\) 1047.77i 0.0404216i
\(162\) 0 0
\(163\) −39790.7 −1.49764 −0.748818 0.662776i \(-0.769378\pi\)
−0.748818 + 0.662776i \(0.769378\pi\)
\(164\) 0 0
\(165\) −13409.2 3530.76i −0.492533 0.129688i
\(166\) 0 0
\(167\) 23773.0 13725.3i 0.852414 0.492141i −0.00905084 0.999959i \(-0.502881\pi\)
0.861465 + 0.507818i \(0.169548\pi\)
\(168\) 0 0
\(169\) 2436.98 4220.97i 0.0853253 0.147788i
\(170\) 0 0
\(171\) 13440.9 + 7605.51i 0.459659 + 0.260097i
\(172\) 0 0
\(173\) 8729.26 + 5039.84i 0.291666 + 0.168393i 0.638693 0.769462i \(-0.279475\pi\)
−0.347027 + 0.937855i \(0.612809\pi\)
\(174\) 0 0
\(175\) 13347.0 + 23117.7i 0.435821 + 0.754864i
\(176\) 0 0
\(177\) −47018.8 + 12817.3i −1.50081 + 0.409120i
\(178\) 0 0
\(179\) 1215.45i 0.0379343i 0.999820 + 0.0189672i \(0.00603779\pi\)
−0.999820 + 0.0189672i \(0.993962\pi\)
\(180\) 0 0
\(181\) 28359.9 0.865661 0.432831 0.901475i \(-0.357515\pi\)
0.432831 + 0.901475i \(0.357515\pi\)
\(182\) 0 0
\(183\) 10850.4 10944.9i 0.324000 0.326821i
\(184\) 0 0
\(185\) 11465.4 6619.55i 0.335001 0.193413i
\(186\) 0 0
\(187\) −3499.68 + 6061.62i −0.100079 + 0.173343i
\(188\) 0 0
\(189\) 50858.0 14338.7i 1.42376 0.401408i
\(190\) 0 0
\(191\) 8445.34 + 4875.92i 0.231500 + 0.133656i 0.611264 0.791427i \(-0.290661\pi\)
−0.379764 + 0.925083i \(0.623995\pi\)
\(192\) 0 0
\(193\) 26701.4 + 46248.2i 0.716836 + 1.24160i 0.962247 + 0.272177i \(0.0877436\pi\)
−0.245411 + 0.969419i \(0.578923\pi\)
\(194\) 0 0
\(195\) −15761.2 15625.2i −0.414497 0.410919i
\(196\) 0 0
\(197\) 68537.7i 1.76603i −0.469349 0.883013i \(-0.655511\pi\)
0.469349 0.883013i \(-0.344489\pi\)
\(198\) 0 0
\(199\) 8237.42 0.208010 0.104005 0.994577i \(-0.466834\pi\)
0.104005 + 0.994577i \(0.466834\pi\)
\(200\) 0 0
\(201\) −10970.0 40242.0i −0.271527 0.996064i
\(202\) 0 0
\(203\) 44961.7 25958.7i 1.09107 0.629927i
\(204\) 0 0
\(205\) 4455.24 7716.70i 0.106014 0.183622i
\(206\) 0 0
\(207\) 1170.83 10.1503i 0.0273246 0.000236884i
\(208\) 0 0
\(209\) −15877.2 9166.71i −0.363481 0.209856i
\(210\) 0 0
\(211\) 20393.7 + 35322.9i 0.458069 + 0.793399i 0.998859 0.0477587i \(-0.0152079\pi\)
−0.540790 + 0.841158i \(0.681875\pi\)
\(212\) 0 0
\(213\) −15347.6 + 58287.4i −0.338284 + 1.28474i
\(214\) 0 0
\(215\) 14303.1i 0.309423i
\(216\) 0 0
\(217\) −21931.2 −0.465740
\(218\) 0 0
\(219\) −41983.7 11054.7i −0.875371 0.230493i
\(220\) 0 0
\(221\) −9701.99 + 5601.45i −0.198644 + 0.114687i
\(222\) 0 0
\(223\) −29959.8 + 51891.9i −0.602461 + 1.04349i 0.389986 + 0.920821i \(0.372480\pi\)
−0.992447 + 0.122672i \(0.960854\pi\)
\(224\) 0 0
\(225\) −25703.6 + 15138.6i −0.507725 + 0.299034i
\(226\) 0 0
\(227\) −85572.3 49405.2i −1.66066 0.958785i −0.972399 0.233325i \(-0.925039\pi\)
−0.688265 0.725459i \(-0.741627\pi\)
\(228\) 0 0
\(229\) 2658.97 + 4605.47i 0.0507040 + 0.0878220i 0.890263 0.455446i \(-0.150520\pi\)
−0.839559 + 0.543268i \(0.817187\pi\)
\(230\) 0 0
\(231\) −60520.3 + 16497.9i −1.13417 + 0.309175i
\(232\) 0 0
\(233\) 18467.8i 0.340176i 0.985429 + 0.170088i \(0.0544052\pi\)
−0.985429 + 0.170088i \(0.945595\pi\)
\(234\) 0 0
\(235\) 63376.3 1.14760
\(236\) 0 0
\(237\) −36299.5 + 36615.5i −0.646254 + 0.651881i
\(238\) 0 0
\(239\) −22383.5 + 12923.1i −0.391861 + 0.226241i −0.682966 0.730450i \(-0.739310\pi\)
0.291105 + 0.956691i \(0.405977\pi\)
\(240\) 0 0
\(241\) −41536.2 + 71942.7i −0.715142 + 1.23866i 0.247763 + 0.968821i \(0.420305\pi\)
−0.962905 + 0.269841i \(0.913029\pi\)
\(242\) 0 0
\(243\) 16515.5 + 56692.4i 0.279691 + 0.960090i
\(244\) 0 0
\(245\) −39586.6 22855.3i −0.659502 0.380764i
\(246\) 0 0
\(247\) −14671.9 25412.4i −0.240487 0.416536i
\(248\) 0 0
\(249\) −18102.2 17945.9i −0.291966 0.289446i
\(250\) 0 0
\(251\) 49051.6i 0.778585i 0.921114 + 0.389292i \(0.127280\pi\)
−0.921114 + 0.389292i \(0.872720\pi\)
\(252\) 0 0
\(253\) −1389.98 −0.0217154
\(254\) 0 0
\(255\) −2760.65 10127.1i −0.0424553 0.155742i
\(256\) 0 0
\(257\) 84096.0 48552.9i 1.27324 0.735104i 0.297641 0.954678i \(-0.403800\pi\)
0.975596 + 0.219574i \(0.0704668\pi\)
\(258\) 0 0
\(259\) 29945.8 51867.7i 0.446413 0.773209i
\(260\) 0 0
\(261\) 29443.1 + 49991.0i 0.432217 + 0.733856i
\(262\) 0 0
\(263\) −35921.0 20739.0i −0.519323 0.299831i 0.217335 0.976097i \(-0.430264\pi\)
−0.736658 + 0.676266i \(0.763597\pi\)
\(264\) 0 0
\(265\) 15757.9 + 27293.5i 0.224391 + 0.388657i
\(266\) 0 0
\(267\) −32719.7 + 124264.i −0.458973 + 1.74310i
\(268\) 0 0
\(269\) 115737.i 1.59944i 0.600370 + 0.799722i \(0.295020\pi\)
−0.600370 + 0.799722i \(0.704980\pi\)
\(270\) 0 0
\(271\) 5077.71 0.0691400 0.0345700 0.999402i \(-0.488994\pi\)
0.0345700 + 0.999402i \(0.488994\pi\)
\(272\) 0 0
\(273\) −97091.7 25565.1i −1.30274 0.343022i
\(274\) 0 0
\(275\) 30668.2 17706.3i 0.405529 0.234133i
\(276\) 0 0
\(277\) 21192.9 36707.2i 0.276205 0.478401i −0.694234 0.719750i \(-0.744257\pi\)
0.970438 + 0.241349i \(0.0775899\pi\)
\(278\) 0 0
\(279\) −212.459 24507.1i −0.00272939 0.314835i
\(280\) 0 0
\(281\) −35902.5 20728.3i −0.454687 0.262513i 0.255121 0.966909i \(-0.417885\pi\)
−0.709807 + 0.704396i \(0.751218\pi\)
\(282\) 0 0
\(283\) 19194.0 + 33245.0i 0.239658 + 0.415100i 0.960616 0.277879i \(-0.0896313\pi\)
−0.720958 + 0.692979i \(0.756298\pi\)
\(284\) 0 0
\(285\) 26525.9 7230.98i 0.326574 0.0890241i
\(286\) 0 0
\(287\) 40309.6i 0.489378i
\(288\) 0 0
\(289\) 78222.5 0.936561
\(290\) 0 0
\(291\) −45401.7 + 45797.0i −0.536149 + 0.540818i
\(292\) 0 0
\(293\) −11239.5 + 6489.12i −0.130922 + 0.0755876i −0.564030 0.825754i \(-0.690750\pi\)
0.433109 + 0.901342i \(0.357417\pi\)
\(294\) 0 0
\(295\) −43380.7 + 75137.6i −0.498486 + 0.863402i
\(296\) 0 0
\(297\) −19021.8 67468.7i −0.215645 0.764873i
\(298\) 0 0
\(299\) −1926.69 1112.37i −0.0215511 0.0124425i
\(300\) 0 0
\(301\) −32352.4 56036.0i −0.357086 0.618492i
\(302\) 0 0
\(303\) −12522.2 12414.1i −0.136394 0.135217i
\(304\) 0 0
\(305\) 27437.4i 0.294947i
\(306\) 0 0
\(307\) 126105. 1.33799 0.668997 0.743265i \(-0.266724\pi\)
0.668997 + 0.743265i \(0.266724\pi\)
\(308\) 0 0
\(309\) −12203.2 44765.8i −0.127807 0.468845i
\(310\) 0 0
\(311\) 86432.4 49901.7i 0.893626 0.515935i 0.0184989 0.999829i \(-0.494111\pi\)
0.875127 + 0.483894i \(0.160778\pi\)
\(312\) 0 0
\(313\) 1112.68 1927.21i 0.0113574 0.0196716i −0.860291 0.509804i \(-0.829718\pi\)
0.871648 + 0.490132i \(0.163051\pi\)
\(314\) 0 0
\(315\) 46327.8 81873.1i 0.466896 0.825125i
\(316\) 0 0
\(317\) −145552. 84034.3i −1.44843 0.836254i −0.450046 0.893005i \(-0.648592\pi\)
−0.998388 + 0.0567515i \(0.981926\pi\)
\(318\) 0 0
\(319\) −34437.0 59646.6i −0.338410 0.586144i
\(320\) 0 0
\(321\) −20893.3 + 79349.1i −0.202767 + 0.770073i
\(322\) 0 0
\(323\) 13878.3i 0.133024i
\(324\) 0 0
\(325\) 56679.9 0.536615
\(326\) 0 0
\(327\) −140655. 37035.8i −1.31541 0.346359i
\(328\) 0 0
\(329\) 248293. 143352.i 2.29389 1.32438i
\(330\) 0 0
\(331\) 3379.43 5853.34i 0.0308452 0.0534254i −0.850191 0.526475i \(-0.823513\pi\)
0.881036 + 0.473049i \(0.156847\pi\)
\(332\) 0 0
\(333\) 58249.6 + 32960.5i 0.525297 + 0.297239i
\(334\) 0 0
\(335\) −64308.1 37128.3i −0.573028 0.330838i
\(336\) 0 0
\(337\) 111046. + 192338.i 0.977787 + 1.69358i 0.670413 + 0.741988i \(0.266117\pi\)
0.307375 + 0.951589i \(0.400550\pi\)
\(338\) 0 0
\(339\) 183208. 49942.5i 1.59421 0.434581i
\(340\) 0 0
\(341\) 29094.2i 0.250206i
\(342\) 0 0
\(343\) −32754.4 −0.278408
\(344\) 0 0
\(345\) 1467.56 1480.34i 0.0123298 0.0124372i
\(346\) 0 0
\(347\) 140770. 81273.8i 1.16910 0.674981i 0.215633 0.976475i \(-0.430819\pi\)
0.953468 + 0.301494i \(0.0974853\pi\)
\(348\) 0 0
\(349\) 36348.5 62957.4i 0.298425 0.516887i −0.677351 0.735660i \(-0.736872\pi\)
0.975776 + 0.218773i \(0.0702054\pi\)
\(350\) 0 0
\(351\) 27627.2 108743.i 0.224245 0.882646i
\(352\) 0 0
\(353\) 111396. + 64314.6i 0.893966 + 0.516131i 0.875238 0.483693i \(-0.160705\pi\)
0.0187281 + 0.999825i \(0.494038\pi\)
\(354\) 0 0
\(355\) 53652.7 + 92929.2i 0.425730 + 0.737387i
\(356\) 0 0
\(357\) −33722.3 33431.2i −0.264594 0.262310i
\(358\) 0 0
\(359\) 95302.8i 0.739463i 0.929139 + 0.369732i \(0.120550\pi\)
−0.929139 + 0.369732i \(0.879450\pi\)
\(360\) 0 0
\(361\) −93969.7 −0.721063
\(362\) 0 0
\(363\) −12769.5 46843.2i −0.0969081 0.355495i
\(364\) 0 0
\(365\) −66935.7 + 38645.4i −0.502426 + 0.290076i
\(366\) 0 0
\(367\) 52073.0 90193.1i 0.386616 0.669639i −0.605376 0.795940i \(-0.706977\pi\)
0.991992 + 0.126301i \(0.0403104\pi\)
\(368\) 0 0
\(369\) 45043.9 390.499i 0.330814 0.00286792i
\(370\) 0 0
\(371\) 123471. + 71286.2i 0.897053 + 0.517914i
\(372\) 0 0
\(373\) −122915. 212896.i −0.883463 1.53020i −0.847465 0.530851i \(-0.821872\pi\)
−0.0359975 0.999352i \(-0.511461\pi\)
\(374\) 0 0
\(375\) −36471.6 + 138513.i −0.259354 + 0.984979i
\(376\) 0 0
\(377\) 110237.i 0.775612i
\(378\) 0 0
\(379\) −200830. −1.39814 −0.699070 0.715053i \(-0.746403\pi\)
−0.699070 + 0.715053i \(0.746403\pi\)
\(380\) 0 0
\(381\) 179819. + 47347.9i 1.23875 + 0.326175i
\(382\) 0 0
\(383\) −117206. + 67668.8i −0.799008 + 0.461308i −0.843124 0.537719i \(-0.819286\pi\)
0.0441160 + 0.999026i \(0.485953\pi\)
\(384\) 0 0
\(385\) −55837.6 + 96713.6i −0.376708 + 0.652478i
\(386\) 0 0
\(387\) 62304.0 36695.0i 0.416001 0.245011i
\(388\) 0 0
\(389\) −175090. 101088.i −1.15708 0.668039i −0.206475 0.978452i \(-0.566199\pi\)
−0.950602 + 0.310413i \(0.899533\pi\)
\(390\) 0 0
\(391\) −526.102 911.236i −0.00344125 0.00596043i
\(392\) 0 0
\(393\) −27798.7 + 7577.92i −0.179986 + 0.0490642i
\(394\) 0 0
\(395\) 91790.2i 0.588304i
\(396\) 0 0
\(397\) 102384. 0.649608 0.324804 0.945781i \(-0.394702\pi\)
0.324804 + 0.945781i \(0.394702\pi\)
\(398\) 0 0
\(399\) 87566.4 88328.8i 0.550037 0.554826i
\(400\) 0 0
\(401\) −78026.4 + 45048.6i −0.485236 + 0.280151i −0.722596 0.691271i \(-0.757051\pi\)
0.237360 + 0.971422i \(0.423718\pi\)
\(402\) 0 0
\(403\) −23283.5 + 40328.2i −0.143363 + 0.248313i
\(404\) 0 0
\(405\) 91937.9 + 50975.9i 0.560512 + 0.310781i
\(406\) 0 0
\(407\) −68808.1 39726.4i −0.415385 0.239823i
\(408\) 0 0
\(409\) −74861.0 129663.i −0.447517 0.775122i 0.550707 0.834699i \(-0.314358\pi\)
−0.998224 + 0.0595770i \(0.981025\pi\)
\(410\) 0 0
\(411\) −24747.3 24533.7i −0.146502 0.145238i
\(412\) 0 0
\(413\) 392495.i 2.30109i
\(414\) 0 0
\(415\) −45379.7 −0.263491
\(416\) 0 0
\(417\) −27645.1 101413.i −0.158981 0.583203i
\(418\) 0 0
\(419\) −109423. + 63175.3i −0.623276 + 0.359848i −0.778143 0.628087i \(-0.783838\pi\)
0.154868 + 0.987935i \(0.450505\pi\)
\(420\) 0 0
\(421\) 107645. 186446.i 0.607335 1.05194i −0.384342 0.923191i \(-0.625572\pi\)
0.991678 0.128745i \(-0.0410949\pi\)
\(422\) 0 0
\(423\) 162594. + 276067.i 0.908709 + 1.54288i
\(424\) 0 0
\(425\) 23215.6 + 13403.5i 0.128529 + 0.0742063i
\(426\) 0 0
\(427\) −62061.3 107493.i −0.340381 0.589557i
\(428\) 0 0
\(429\) −33914.9 + 128803.i −0.184279 + 0.699859i
\(430\) 0 0
\(431\) 121972.i 0.656607i −0.944572 0.328304i \(-0.893523\pi\)
0.944572 0.328304i \(-0.106477\pi\)
\(432\) 0 0
\(433\) −152419. −0.812951 −0.406475 0.913662i \(-0.633242\pi\)
−0.406475 + 0.913662i \(0.633242\pi\)
\(434\) 0 0
\(435\) 99882.9 + 26300.0i 0.527853 + 0.138988i
\(436\) 0 0
\(437\) 2386.80 1378.02i 0.0124984 0.00721593i
\(438\) 0 0
\(439\) 115564. 200162.i 0.599643 1.03861i −0.393230 0.919440i \(-0.628642\pi\)
0.992874 0.119173i \(-0.0380242\pi\)
\(440\) 0 0
\(441\) −2003.26 231075.i −0.0103005 1.18816i
\(442\) 0 0
\(443\) 17481.4 + 10092.9i 0.0890774 + 0.0514289i 0.543877 0.839165i \(-0.316956\pi\)
−0.454800 + 0.890594i \(0.650289\pi\)
\(444\) 0 0
\(445\) 114383. + 198117.i 0.577618 + 1.00046i
\(446\) 0 0
\(447\) −131044. + 35722.7i −0.655847 + 0.178784i
\(448\) 0 0
\(449\) 92962.8i 0.461123i 0.973058 + 0.230561i \(0.0740562\pi\)
−0.973058 + 0.230561i \(0.925944\pi\)
\(450\) 0 0
\(451\) −53475.0 −0.262905
\(452\) 0 0
\(453\) 191707. 193376.i 0.934205 0.942339i
\(454\) 0 0
\(455\) −154796. + 89371.5i −0.747716 + 0.431694i
\(456\) 0 0
\(457\) −110443. + 191293.i −0.528819 + 0.915941i 0.470616 + 0.882338i \(0.344032\pi\)
−0.999435 + 0.0336033i \(0.989302\pi\)
\(458\) 0 0
\(459\) 37031.0 38006.9i 0.175768 0.180400i
\(460\) 0 0
\(461\) 91967.2 + 53097.3i 0.432744 + 0.249845i 0.700515 0.713638i \(-0.252954\pi\)
−0.267771 + 0.963483i \(0.586287\pi\)
\(462\) 0 0
\(463\) −34090.0 59045.5i −0.159025 0.275439i 0.775493 0.631357i \(-0.217502\pi\)
−0.934517 + 0.355918i \(0.884168\pi\)
\(464\) 0 0
\(465\) −30985.4 30718.0i −0.143302 0.142065i
\(466\) 0 0
\(467\) 224350.i 1.02871i 0.857578 + 0.514353i \(0.171968\pi\)
−0.857578 + 0.514353i \(0.828032\pi\)
\(468\) 0 0
\(469\) −335925. −1.52720
\(470\) 0 0
\(471\) −48814.7 179070.i −0.220043 0.807202i
\(472\) 0 0
\(473\) −74337.9 + 42919.0i −0.332267 + 0.191835i
\(474\) 0 0
\(475\) −35107.8 + 60808.5i −0.155603 + 0.269511i
\(476\) 0 0
\(477\) −78462.7 + 138664.i −0.344847 + 0.609433i
\(478\) 0 0
\(479\) 94542.7 + 54584.3i 0.412057 + 0.237901i 0.691673 0.722211i \(-0.256874\pi\)
−0.279616 + 0.960112i \(0.590207\pi\)
\(480\) 0 0
\(481\) −63584.5 110132.i −0.274828 0.476016i
\(482\) 0 0
\(483\) 2401.14 9119.10i 0.0102926 0.0390893i
\(484\) 0 0
\(485\) 114807.i 0.488073i
\(486\) 0 0
\(487\) −106309. −0.448240 −0.224120 0.974562i \(-0.571951\pi\)
−0.224120 + 0.974562i \(0.571951\pi\)
\(488\) 0 0
\(489\) −346312. 91187.0i −1.44827 0.381343i
\(490\) 0 0
\(491\) 64472.7 37223.3i 0.267432 0.154402i −0.360288 0.932841i \(-0.617322\pi\)
0.627720 + 0.778439i \(0.283988\pi\)
\(492\) 0 0
\(493\) 26068.5 45152.0i 0.107256 0.185773i
\(494\) 0 0
\(495\) −108614. 61458.9i −0.443275 0.250827i
\(496\) 0 0
\(497\) 420397. + 242716.i 1.70195 + 0.982621i
\(498\) 0 0
\(499\) −55603.4 96308.0i −0.223306 0.386777i 0.732504 0.680763i \(-0.238352\pi\)
−0.955810 + 0.293985i \(0.905018\pi\)
\(500\) 0 0
\(501\) 238358. 64976.6i 0.949631 0.258870i
\(502\) 0 0
\(503\) 115897.i 0.458077i 0.973417 + 0.229038i \(0.0735581\pi\)
−0.973417 + 0.229038i \(0.926442\pi\)
\(504\) 0 0
\(505\) −31391.5 −0.123092
\(506\) 0 0
\(507\) 30882.9 31151.8i 0.120144 0.121190i
\(508\) 0 0
\(509\) −170299. + 98322.1i −0.657319 + 0.379503i −0.791255 0.611487i \(-0.790572\pi\)
0.133936 + 0.990990i \(0.457238\pi\)
\(510\) 0 0
\(511\) −174825. + 302807.i −0.669519 + 1.15964i
\(512\) 0 0
\(513\) 99551.4 + 96995.4i 0.378279 + 0.368567i
\(514\) 0 0
\(515\) −71537.3 41302.1i −0.269723 0.155725i
\(516\) 0 0
\(517\) −190172. 329388.i −0.711486 1.23233i
\(518\) 0 0
\(519\) 64424.2 + 63868.1i 0.239174 + 0.237110i
\(520\) 0 0
\(521\) 299306.i 1.10266i −0.834289 0.551328i \(-0.814121\pi\)
0.834289 0.551328i \(-0.185879\pi\)
\(522\) 0 0
\(523\) 162892. 0.595520 0.297760 0.954641i \(-0.403760\pi\)
0.297760 + 0.954641i \(0.403760\pi\)
\(524\) 0 0
\(525\) 63185.6 + 231788.i 0.229245 + 0.840955i
\(526\) 0 0
\(527\) −19073.4 + 11012.0i −0.0686763 + 0.0396503i
\(528\) 0 0
\(529\) −139816. + 242168.i −0.499627 + 0.865379i
\(530\) 0 0
\(531\) −438594. + 3802.29i −1.55551 + 0.0134852i
\(532\) 0 0
\(533\) −74123.1 42795.0i −0.260915 0.150639i
\(534\) 0 0
\(535\) 73039.7 + 126508.i 0.255183 + 0.441989i
\(536\) 0 0
\(537\) −2785.41 + 10578.5i −0.00965920 + 0.0366839i
\(538\) 0 0
\(539\) 274327.i 0.944257i
\(540\) 0 0
\(541\) −62918.7 −0.214974 −0.107487 0.994207i \(-0.534280\pi\)
−0.107487 + 0.994207i \(0.534280\pi\)
\(542\) 0 0
\(543\) 246826. + 64991.5i 0.837128 + 0.220423i
\(544\) 0 0
\(545\) −224251. + 129471.i −0.754989 + 0.435893i
\(546\) 0 0
\(547\) −7403.57 + 12823.4i −0.0247438 + 0.0428575i −0.878132 0.478418i \(-0.841210\pi\)
0.853388 + 0.521276i \(0.174544\pi\)
\(548\) 0 0
\(549\) 119517. 70391.8i 0.396539 0.233549i
\(550\) 0 0
\(551\) 118267. + 68281.3i 0.389547 + 0.224905i
\(552\) 0 0
\(553\) 207622. + 359612.i 0.678927 + 1.17594i
\(554\) 0 0
\(555\) 114957. 31337.4i 0.373207 0.101736i
\(556\) 0 0
\(557\) 254392.i 0.819962i −0.912094 0.409981i \(-0.865535\pi\)
0.912094 0.409981i \(-0.134465\pi\)
\(558\) 0 0
\(559\) −137389. −0.439671
\(560\) 0 0
\(561\) −44350.2 + 44736.3i −0.140919 + 0.142146i
\(562\) 0 0
\(563\) −80806.7 + 46653.8i −0.254936 + 0.147187i −0.622022 0.783000i \(-0.713689\pi\)
0.367087 + 0.930187i \(0.380355\pi\)
\(564\) 0 0
\(565\) 169032. 292772.i 0.529508 0.917134i
\(566\) 0 0
\(567\) 475494. 8245.01i 1.47904 0.0256463i
\(568\) 0 0
\(569\) −136853. 79012.1i −0.422698 0.244045i 0.273533 0.961863i \(-0.411808\pi\)
−0.696231 + 0.717818i \(0.745141\pi\)
\(570\) 0 0
\(571\) −219188. 379644.i −0.672270 1.16441i −0.977259 0.212050i \(-0.931986\pi\)
0.304988 0.952356i \(-0.401347\pi\)
\(572\) 0 0
\(573\) 62328.8 + 61790.8i 0.189836 + 0.188198i
\(574\) 0 0
\(575\) 5323.52i 0.0161014i
\(576\) 0 0
\(577\) 214770. 0.645094 0.322547 0.946554i \(-0.395461\pi\)
0.322547 + 0.946554i \(0.395461\pi\)
\(578\) 0 0
\(579\) 126406. + 463705.i 0.377061 + 1.38320i
\(580\) 0 0
\(581\) −177787. + 102645.i −0.526681 + 0.304079i
\(582\) 0 0
\(583\) 94568.9 163798.i 0.278235 0.481917i
\(584\) 0 0
\(585\) −101368. 172111.i −0.296202 0.502918i
\(586\) 0 0
\(587\) −568470. 328206.i −1.64980 0.952512i −0.977150 0.212550i \(-0.931823\pi\)
−0.672649 0.739962i \(-0.734843\pi\)
\(588\) 0 0
\(589\) −28843.8 49959.0i −0.0831423 0.144007i
\(590\) 0 0
\(591\) 157066. 596507.i 0.449683 1.70781i
\(592\) 0 0
\(593\) 118369.i 0.336612i 0.985735 + 0.168306i \(0.0538296\pi\)
−0.985735 + 0.168306i \(0.946170\pi\)
\(594\) 0 0
\(595\) −84537.3 −0.238789
\(596\) 0 0
\(597\) 71693.1 + 18877.4i 0.201154 + 0.0529656i
\(598\) 0 0
\(599\) 306693. 177069.i 0.854772 0.493503i −0.00748620 0.999972i \(-0.502383\pi\)
0.862258 + 0.506469i \(0.169050\pi\)
\(600\) 0 0
\(601\) 297820. 515839.i 0.824526 1.42812i −0.0777554 0.996972i \(-0.524775\pi\)
0.902281 0.431148i \(-0.141891\pi\)
\(602\) 0 0
\(603\) −3254.27 375379.i −0.00894992 1.03237i
\(604\) 0 0
\(605\) −74857.1 43218.8i −0.204514 0.118076i
\(606\) 0 0
\(607\) 4860.81 + 8419.17i 0.0131926 + 0.0228503i 0.872546 0.488531i \(-0.162467\pi\)
−0.859354 + 0.511382i \(0.829134\pi\)
\(608\) 0 0
\(609\) 450806. 122890.i 1.21550 0.331346i
\(610\) 0 0
\(611\) 608765.i 1.63067i
\(612\) 0 0
\(613\) 194450. 0.517473 0.258736 0.965948i \(-0.416694\pi\)
0.258736 + 0.965948i \(0.416694\pi\)
\(614\) 0 0
\(615\) 56459.6 56951.2i 0.149275 0.150575i
\(616\) 0 0
\(617\) 422308. 243819.i 1.10932 0.640469i 0.170670 0.985328i \(-0.445407\pi\)
0.938654 + 0.344860i \(0.112073\pi\)
\(618\) 0 0
\(619\) 8628.23 14944.5i 0.0225185 0.0390033i −0.854547 0.519375i \(-0.826165\pi\)
0.877065 + 0.480372i \(0.159498\pi\)
\(620\) 0 0
\(621\) 10213.4 + 2594.81i 0.0264842 + 0.00672857i
\(622\) 0 0
\(623\) 896249. + 517449.i 2.30915 + 1.33319i
\(624\) 0 0
\(625\) 12412.5 + 21499.1i 0.0317760 + 0.0550376i
\(626\) 0 0
\(627\) −117178. 116166.i −0.298065 0.295492i
\(628\) 0 0
\(629\) 60145.1i 0.152019i
\(630\) 0 0
\(631\) 429836. 1.07955 0.539777 0.841808i \(-0.318509\pi\)
0.539777 + 0.841808i \(0.318509\pi\)
\(632\) 0 0
\(633\) 96545.0 + 354163.i 0.240948 + 0.883886i
\(634\) 0 0
\(635\) 286690. 165521.i 0.710993 0.410492i
\(636\) 0 0
\(637\) −219538. + 380251.i −0.541042 + 0.937113i
\(638\) 0 0
\(639\) −267151. + 472124.i −0.654266 + 1.15626i
\(640\) 0 0
\(641\) −27979.5 16154.0i −0.0680964 0.0393155i 0.465565 0.885014i \(-0.345851\pi\)
−0.533662 + 0.845698i \(0.679184\pi\)
\(642\) 0 0
\(643\) 138524. + 239931.i 0.335045 + 0.580315i 0.983494 0.180943i \(-0.0579150\pi\)
−0.648448 + 0.761259i \(0.724582\pi\)
\(644\) 0 0
\(645\) 32777.9 124485.i 0.0787882 0.299224i
\(646\) 0 0
\(647\) 125452.i 0.299689i −0.988710 0.149844i \(-0.952123\pi\)
0.988710 0.149844i \(-0.0478773\pi\)
\(648\) 0 0
\(649\) 520687. 1.23620
\(650\) 0 0
\(651\) −190875. 50259.1i −0.450388 0.118591i
\(652\) 0 0
\(653\) 245936. 141991.i 0.576760 0.332992i −0.183085 0.983097i \(-0.558608\pi\)
0.759845 + 0.650105i \(0.225275\pi\)
\(654\) 0 0
\(655\) −25647.8 + 44423.2i −0.0597815 + 0.103545i
\(656\) 0 0
\(657\) −340065. 192425.i −0.787828 0.445791i
\(658\) 0 0
\(659\) −311139. 179636.i −0.716446 0.413640i 0.0969973 0.995285i \(-0.469076\pi\)
−0.813443 + 0.581644i \(0.802410\pi\)
\(660\) 0 0
\(661\) −98902.1 171303.i −0.226362 0.392070i 0.730366 0.683057i \(-0.239350\pi\)
−0.956727 + 0.290987i \(0.906016\pi\)
\(662\) 0 0
\(663\) −97276.5 + 26517.6i −0.221300 + 0.0603264i
\(664\) 0 0
\(665\) 221429.i 0.500715i
\(666\) 0 0
\(667\) 10353.7 0.0232726
\(668\) 0 0
\(669\) −379669. + 382975.i −0.848307 + 0.855693i
\(670\) 0 0
\(671\) −142602. + 82331.1i −0.316723 + 0.182860i
\(672\) 0 0
\(673\) −119036. + 206176.i −0.262814 + 0.455207i −0.966989 0.254820i \(-0.917984\pi\)
0.704175 + 0.710027i \(0.251317\pi\)
\(674\) 0 0
\(675\) −258400. + 72852.2i −0.567133 + 0.159895i
\(676\) 0 0
\(677\) 216636. + 125075.i 0.472666 + 0.272894i 0.717355 0.696708i \(-0.245353\pi\)
−0.244689 + 0.969602i \(0.578686\pi\)
\(678\) 0 0
\(679\) 259684. + 449786.i 0.563256 + 0.975588i
\(680\) 0 0
\(681\) −631545. 626094.i −1.36179 1.35004i
\(682\) 0 0
\(683\) 267358.i 0.573129i 0.958061 + 0.286564i \(0.0925132\pi\)
−0.958061 + 0.286564i \(0.907487\pi\)
\(684\) 0 0
\(685\) −62038.2 −0.132214
\(686\) 0 0
\(687\) 12587.7 + 46176.5i 0.0266707 + 0.0978380i
\(688\) 0 0
\(689\) 262169. 151363.i 0.552259 0.318847i
\(690\) 0 0
\(691\) 168343. 291579.i 0.352565 0.610660i −0.634133 0.773224i \(-0.718643\pi\)
0.986698 + 0.162564i \(0.0519763\pi\)
\(692\) 0 0
\(693\) −564537. + 4894.13i −1.17551 + 0.0101908i
\(694\) 0 0
\(695\) −162061. 93565.9i −0.335513 0.193708i
\(696\) 0 0
\(697\) −20240.1 35056.9i −0.0416627 0.0721618i
\(698\) 0 0
\(699\) −42322.1 + 160732.i −0.0866189 + 0.328963i
\(700\) 0 0
\(701\) 635795.i 1.29384i −0.762557 0.646921i \(-0.776056\pi\)
0.762557 0.646921i \(-0.223944\pi\)
\(702\) 0 0
\(703\) 157538. 0.318768
\(704\) 0 0
\(705\) 551586. + 145237.i 1.10978 + 0.292214i
\(706\) 0 0
\(707\) −122984. + 71005.0i −0.246043 + 0.142053i
\(708\) 0 0
\(709\) −215513. + 373280.i −0.428728 + 0.742578i −0.996760 0.0804279i \(-0.974371\pi\)
0.568033 + 0.823006i \(0.307705\pi\)
\(710\) 0 0
\(711\) −399837. + 235491.i −0.790941 + 0.465839i
\(712\) 0 0
\(713\) −3787.73 2186.84i −0.00745074 0.00430169i
\(714\) 0 0
\(715\) 118561. + 205354.i 0.231916 + 0.401690i
\(716\) 0 0
\(717\) −224427. + 61178.9i −0.436553 + 0.119004i
\(718\) 0 0
\(719\) 798370.i 1.54435i −0.635409 0.772176i \(-0.719168\pi\)
0.635409 0.772176i \(-0.280832\pi\)
\(720\) 0 0
\(721\) −373688. −0.718850
\(722\) 0 0
\(723\) −526372. + 530956.i −1.00697 + 1.01574i
\(724\) 0 0
\(725\) −228442. + 131891.i −0.434610 + 0.250922i
\(726\) 0 0
\(727\) 158557. 274628.i 0.299996 0.519609i −0.676139 0.736775i \(-0.736348\pi\)
0.976135 + 0.217166i \(0.0696812\pi\)
\(728\) 0 0
\(729\) 13819.7 + 531261.i 0.0260043 + 0.999662i
\(730\) 0 0
\(731\) −56273.2 32489.3i −0.105309 0.0608004i
\(732\) 0 0
\(733\) 488922. + 846837.i 0.909979 + 1.57613i 0.814091 + 0.580738i \(0.197236\pi\)
0.0958883 + 0.995392i \(0.469431\pi\)
\(734\) 0 0
\(735\) −292159. 289637.i −0.540810 0.536142i
\(736\) 0 0
\(737\) 445641.i 0.820446i
\(738\) 0 0
\(739\) 392565. 0.718825 0.359412 0.933179i \(-0.382977\pi\)
0.359412 + 0.933179i \(0.382977\pi\)
\(740\) 0 0
\(741\) −69457.6 254796.i −0.126498 0.464041i
\(742\) 0 0
\(743\) −899450. + 519298.i −1.62929 + 0.940673i −0.644990 + 0.764191i \(0.723139\pi\)
−0.984304 + 0.176482i \(0.943528\pi\)
\(744\) 0 0
\(745\) −120905. + 209413.i −0.217836 + 0.377304i
\(746\) 0 0
\(747\) −116423. 197674.i −0.208641 0.354248i
\(748\) 0 0
\(749\) 572304. + 330420.i 1.02015 + 0.588983i
\(750\) 0 0
\(751\) 159783. + 276752.i 0.283302 + 0.490694i 0.972196 0.234168i \(-0.0752367\pi\)
−0.688894 + 0.724862i \(0.741903\pi\)
\(752\) 0 0
\(753\) −112410. + 426913.i −0.198251 + 0.752921i
\(754\) 0 0
\(755\) 484769.i 0.850435i
\(756\) 0 0
\(757\) 500321. 0.873085 0.436543 0.899684i \(-0.356203\pi\)
0.436543 + 0.899684i \(0.356203\pi\)
\(758\) 0 0
\(759\) −12097.5 3185.37i −0.0209996 0.00552939i
\(760\) 0 0
\(761\) −783645. + 452437.i −1.35316 + 0.781249i −0.988691 0.149966i \(-0.952083\pi\)
−0.364471 + 0.931215i \(0.618750\pi\)
\(762\) 0 0
\(763\) −585707. + 1.01447e6i −1.00608 + 1.74258i
\(764\) 0 0
\(765\) −818.955 94466.2i −0.00139938 0.161419i
\(766\) 0 0
\(767\) 721738. + 416696.i 1.22684 + 0.708318i
\(768\) 0 0
\(769\) −71688.6 124168.i −0.121227 0.209970i 0.799025 0.601298i \(-0.205349\pi\)
−0.920252 + 0.391327i \(0.872016\pi\)
\(770\) 0 0
\(771\) 843184. 229852.i 1.41845 0.386670i
\(772\) 0 0
\(773\) 710063.i 1.18833i −0.804342 0.594166i \(-0.797482\pi\)
0.804342 0.594166i \(-0.202518\pi\)
\(774\) 0 0
\(775\) 111428. 0.185521
\(776\) 0 0
\(777\) 379492. 382796.i 0.628580 0.634053i
\(778\) 0 0
\(779\) 91824.5 53014.9i 0.151316 0.0873621i
\(780\) 0 0
\(781\) 321990. 557702.i 0.527885 0.914324i
\(782\) 0 0
\(783\) 141691. + 502563.i 0.231109 + 0.819723i
\(784\) 0 0
\(785\) −286161. 165215.i −0.464377 0.268108i
\(786\) 0 0
\(787\) −300855. 521097.i −0.485745 0.841335i 0.514121 0.857718i \(-0.328118\pi\)
−0.999866 + 0.0163827i \(0.994785\pi\)
\(788\) 0 0
\(789\) −265106. 262818.i −0.425859 0.422183i
\(790\) 0 0
\(791\) 1.52935e6i 2.44429i
\(792\) 0 0
\(793\) −263552. −0.419102
\(794\) 0 0
\(795\) 74598.8 + 273656.i 0.118031 + 0.432983i
\(796\) 0 0
\(797\) −372227. + 214905.i −0.585991 + 0.338322i −0.763511 0.645795i \(-0.776526\pi\)
0.177519 + 0.984117i \(0.443193\pi\)
\(798\) 0 0
\(799\) 143959. 249344.i 0.225499 0.390576i
\(800\) 0 0
\(801\) −569542. + 1.00653e6i −0.887688 + 1.56877i
\(802\) 0 0
\(803\) 401706. + 231925.i 0.622984 + 0.359680i
\(804\) 0 0
\(805\) −8393.99 14538.8i −0.0129532 0.0224356i
\(806\) 0 0
\(807\) −265232. + 1.00730e6i −0.407266 + 1.54672i
\(808\) 0 0
\(809\) 629639.i 0.962042i −0.876709 0.481021i \(-0.840266\pi\)
0.876709 0.481021i \(-0.159734\pi\)
\(810\) 0 0
\(811\) −577760. −0.878428 −0.439214 0.898383i \(-0.644743\pi\)
−0.439214 + 0.898383i \(0.644743\pi\)
\(812\) 0 0
\(813\) 44193.1 + 11636.4i 0.0668610 + 0.0176051i
\(814\) 0 0
\(815\) −552135. + 318775.i −0.831246 + 0.479920i
\(816\) 0 0
\(817\) 85099.3 147396.i 0.127492 0.220822i
\(818\) 0 0
\(819\) −786436. 445004.i −1.17245 0.663431i
\(820\) 0 0
\(821\) 38728.3 + 22359.8i 0.0574569 + 0.0331727i 0.528453 0.848962i \(-0.322772\pi\)
−0.470996 + 0.882135i \(0.656106\pi\)
\(822\) 0 0
\(823\) −146289. 253380.i −0.215979 0.374087i 0.737596 0.675242i \(-0.235961\pi\)
−0.953575 + 0.301155i \(0.902628\pi\)
\(824\) 0 0
\(825\) 307493. 83822.6i 0.451780 0.123155i
\(826\) 0 0
\(827\) 806491.i 1.17920i 0.807694 + 0.589601i \(0.200715\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(828\) 0 0
\(829\) 204904. 0.298154 0.149077 0.988826i \(-0.452370\pi\)
0.149077 + 0.988826i \(0.452370\pi\)
\(830\) 0 0
\(831\) 268570. 270908.i 0.388916 0.392302i
\(832\) 0 0
\(833\) −179842. + 103832.i −0.259179 + 0.149637i
\(834\) 0 0
\(835\) 219915. 380905.i 0.315415 0.546315i
\(836\) 0 0
\(837\) 54313.0 213780.i 0.0775269 0.305152i
\(838\) 0 0
\(839\) −667796. 385552.i −0.948681 0.547721i −0.0560100 0.998430i \(-0.517838\pi\)
−0.892671 + 0.450709i \(0.851171\pi\)
\(840\) 0 0
\(841\) −97125.1 168226.i −0.137322 0.237848i
\(842\) 0 0
\(843\) −264970. 262682.i −0.372856 0.369637i
\(844\) 0 0
\(845\) 78093.4i 0.109371i
\(846\) 0 0
\(847\) −391030. −0.545058
\(848\) 0 0
\(849\) 90865.5 + 333329.i 0.126062 + 0.462442i
\(850\) 0 0
\(851\) 10343.8 5972.02i 0.0142831 0.00824635i
\(852\) 0 0
\(853\) −541184. + 937358.i −0.743784 + 1.28827i 0.206977 + 0.978346i \(0.433637\pi\)
−0.950761 + 0.309925i \(0.899696\pi\)
\(854\) 0 0
\(855\) 247435. 2145.09i 0.338477 0.00293436i
\(856\) 0 0
\(857\) −461890. 266672.i −0.628893 0.363092i 0.151430 0.988468i \(-0.451612\pi\)
−0.780323 + 0.625376i \(0.784945\pi\)
\(858\) 0 0
\(859\) 565525. + 979517.i 0.766417 + 1.32747i 0.939494 + 0.342565i \(0.111296\pi\)
−0.173077 + 0.984908i \(0.555371\pi\)
\(860\) 0 0
\(861\) 92376.1 350828.i 0.124610 0.473247i
\(862\) 0 0
\(863\) 110025.i 0.147730i −0.997268 0.0738649i \(-0.976467\pi\)
0.997268 0.0738649i \(-0.0235334\pi\)
\(864\) 0 0
\(865\) 161503. 0.215848
\(866\) 0 0
\(867\) 680798. + 179260.i 0.905691 + 0.238476i
\(868\) 0 0
\(869\) 477065. 275433.i 0.631739 0.364735i
\(870\) 0 0
\(871\) −356638. + 617715.i −0.470101 + 0.814238i
\(872\) 0 0
\(873\) −500098. + 294541.i −0.656185 + 0.386472i
\(874\) 0 0
\(875\) 999020. + 576784.i 1.30484 + 0.753351i
\(876\) 0 0
\(877\) 204377. + 353991.i 0.265725 + 0.460249i 0.967753 0.251900i \(-0.0810554\pi\)
−0.702028 + 0.712149i \(0.747722\pi\)
\(878\) 0 0
\(879\) −112692. + 30719.9i −0.145853 + 0.0397596i
\(880\) 0 0
\(881\) 235809.i 0.303815i −0.988395 0.151907i \(-0.951458\pi\)
0.988395 0.151907i \(-0.0485416\pi\)
\(882\) 0 0
\(883\) 273719. 0.351061 0.175531 0.984474i \(-0.443836\pi\)
0.175531 + 0.984474i \(0.443836\pi\)
\(884\) 0 0
\(885\) −549748. + 554534.i −0.701903 + 0.708014i
\(886\) 0 0
\(887\) −322071. + 185948.i −0.409359 + 0.236344i −0.690514 0.723319i \(-0.742616\pi\)
0.281155 + 0.959662i \(0.409282\pi\)
\(888\) 0 0
\(889\) 748789. 1.29694e6i 0.947449 1.64103i
\(890\) 0 0
\(891\) −10937.9 630795.i −0.0137778 0.794571i
\(892\) 0 0
\(893\) 653108. + 377072.i 0.818996 + 0.472848i
\(894\) 0 0
\(895\) 9737.36 + 16865.6i 0.0121561 + 0.0210550i
\(896\) 0 0
\(897\) −14219.4 14096.7i −0.0176725 0.0175199i
\(898\) 0 0
\(899\) 216718.i 0.268148i
\(900\) 0 0
\(901\) 143176. 0.176368
\(902\) 0 0
\(903\) −153158. 561842.i −0.187830 0.689030i
\(904\) 0 0
\(905\) 393522. 227200.i 0.480476 0.277403i
\(906\) 0 0
\(907\) −71965.4 + 124648.i −0.0874801 + 0.151520i −0.906445 0.422323i \(-0.861215\pi\)
0.818965 + 0.573843i \(0.194548\pi\)
\(908\) 0 0
\(909\) −80536.0 136741.i −0.0974680 0.165490i
\(910\) 0 0
\(911\) 772201. + 445830.i 0.930451 + 0.537196i 0.886954 0.461857i \(-0.152817\pi\)
0.0434969 + 0.999054i \(0.486150\pi\)
\(912\) 0 0
\(913\) 136170. + 235854.i 0.163358 + 0.282944i
\(914\) 0 0
\(915\) 62877.5 238798.i 0.0751023 0.285225i
\(916\) 0 0
\(917\) 232053.i 0.275961i
\(918\) 0 0
\(919\) 582156. 0.689300 0.344650 0.938731i \(-0.387998\pi\)
0.344650 + 0.938731i \(0.387998\pi\)
\(920\) 0 0
\(921\) 1.09753e6 + 288990.i 1.29389 + 0.340693i
\(922\) 0 0
\(923\) 892636. 515364.i 1.04778 0.604937i
\(924\) 0 0
\(925\) −152149. + 263530.i −0.177822 + 0.307997i
\(926\) 0 0
\(927\) −3620.10 417578.i −0.00421271 0.485935i
\(928\) 0 0
\(929\) −668027. 385685.i −0.774038 0.446891i 0.0602751 0.998182i \(-0.480802\pi\)
−0.834313 + 0.551291i \(0.814136\pi\)
\(930\) 0 0
\(931\) −271966. 471059.i −0.313773 0.543470i
\(932\) 0 0
\(933\) 866609. 236238.i 0.995543 0.271385i
\(934\) 0 0
\(935\) 112148.i 0.128283i
\(936\) 0 0
\(937\) −1.46097e6 −1.66403 −0.832015 0.554753i \(-0.812813\pi\)
−0.832015 + 0.554753i \(0.812813\pi\)
\(938\) 0 0
\(939\) 14100.5 14223.3i 0.0159921 0.0161313i
\(940\) 0 0
\(941\) 146658. 84673.2i 0.165626 0.0956240i −0.414896 0.909869i \(-0.636182\pi\)
0.580522 + 0.814245i \(0.302849\pi\)
\(942\) 0 0
\(943\) 4019.42 6961.83i 0.00452001 0.00782889i
\(944\) 0 0
\(945\) 590833. 606402.i 0.661608 0.679042i
\(946\) 0 0
\(947\) −514322. 296944.i −0.573503 0.331112i 0.185044 0.982730i \(-0.440757\pi\)
−0.758547 + 0.651618i \(0.774090\pi\)
\(948\) 0 0
\(949\) 371210. + 642955.i 0.412180 + 0.713917i
\(950\) 0 0
\(951\) −1.07421e6 1.06494e6i −1.18776 1.17750i
\(952\) 0 0
\(953\) 1.25477e6i 1.38159i 0.723050 + 0.690796i \(0.242740\pi\)
−0.723050 + 0.690796i \(0.757260\pi\)
\(954\) 0 0
\(955\) 156250. 0.171322
\(956\) 0 0
\(957\) −163027. 598043.i −0.178006 0.652993i
\(958\) 0 0
\(959\) −243051. + 140326.i −0.264278 + 0.152581i
\(960\) 0 0
\(961\) 415987. 720510.i 0.450436 0.780178i
\(962\) 0 0
\(963\) −363684. + 642722.i −0.392167 + 0.693060i
\(964\) 0 0
\(965\) 741017. + 427826.i 0.795744 + 0.459423i
\(966\) 0 0
\(967\) −528928. 916130.i −0.565645 0.979725i −0.996989 0.0775381i \(-0.975294\pi\)
0.431345 0.902187i \(-0.358039\pi\)
\(968\) 0 0
\(969\) 31804.4 120787.i 0.0338719 0.128639i
\(970\) 0 0
\(971\) 775745.i 0.822774i 0.911461 + 0.411387i \(0.134956\pi\)
−0.911461 + 0.411387i \(0.865044\pi\)
\(972\) 0 0
\(973\) −846555. −0.894189
\(974\) 0 0
\(975\) 493305. + 129892.i 0.518927 + 0.136638i
\(976\) 0 0
\(977\) −429995. + 248258.i −0.450479 + 0.260084i −0.708032 0.706180i \(-0.750417\pi\)
0.257554 + 0.966264i \(0.417084\pi\)
\(978\) 0 0
\(979\) 686453. 1.18897e6i 0.716218 1.24053i
\(980\) 0 0
\(981\) −1.13930e6 644671.i −1.18386 0.669885i
\(982\) 0 0
\(983\) −260643. 150482.i −0.269736 0.155732i 0.359032 0.933325i \(-0.383107\pi\)
−0.628768 + 0.777593i \(0.716440\pi\)
\(984\) 0 0
\(985\) −549076. 951028.i −0.565927 0.980214i
\(986\) 0 0
\(987\) 2.48950e6 678638.i 2.55551 0.696633i
\(988\) 0 0
\(989\) 12903.9i 0.0131925i
\(990\) 0 0
\(991\) −1.70094e6 −1.73197 −0.865987 0.500067i \(-0.833309\pi\)
−0.865987 + 0.500067i \(0.833309\pi\)
\(992\) 0 0
\(993\) 42826.3 43199.1i 0.0434322 0.0438103i
\(994\) 0 0
\(995\) 114302. 65992.4i 0.115454 0.0666573i
\(996\) 0 0
\(997\) 312919. 541991.i 0.314805 0.545258i −0.664591 0.747207i \(-0.731394\pi\)
0.979396 + 0.201949i \(0.0647276\pi\)
\(998\) 0 0
\(999\) 431432. + 420355.i 0.432297 + 0.421197i
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 144.5.q.a.65.3 6
3.2 odd 2 432.5.q.a.305.1 6
4.3 odd 2 9.5.d.a.2.3 6
9.2 odd 6 1296.5.e.c.161.2 6
9.4 even 3 432.5.q.a.17.1 6
9.5 odd 6 inner 144.5.q.a.113.3 6
9.7 even 3 1296.5.e.c.161.5 6
12.11 even 2 27.5.d.a.8.1 6
36.7 odd 6 81.5.b.a.80.2 6
36.11 even 6 81.5.b.a.80.5 6
36.23 even 6 9.5.d.a.5.3 yes 6
36.31 odd 6 27.5.d.a.17.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.5.d.a.2.3 6 4.3 odd 2
9.5.d.a.5.3 yes 6 36.23 even 6
27.5.d.a.8.1 6 12.11 even 2
27.5.d.a.17.1 6 36.31 odd 6
81.5.b.a.80.2 6 36.7 odd 6
81.5.b.a.80.5 6 36.11 even 6
144.5.q.a.65.3 6 1.1 even 1 trivial
144.5.q.a.113.3 6 9.5 odd 6 inner
432.5.q.a.17.1 6 9.4 even 3
432.5.q.a.305.1 6 3.2 odd 2
1296.5.e.c.161.2 6 9.2 odd 6
1296.5.e.c.161.5 6 9.7 even 3