Properties

Label 208.4.w.e.49.2
Level $208$
Weight $4$
Character 208.49
Analytic conductor $12.272$
Analytic rank $0$
Dimension $20$
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] = [208,4,Mod(17,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.w (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: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 340 x^{18} + 48278 x^{16} + 3724852 x^{14} + 170209937 x^{12} + 4703455168 x^{10} + 76591569568 x^{8} + 674465272704 x^{6} + \cdots + 549543481344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{40} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.2
Root \(-7.81696i\) of defining polynomial
Character \(\chi\) \(=\) 208.49
Dual form 208.4.w.e.17.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.40848 - 5.90366i) q^{3} +12.7779i q^{5} +(2.31195 + 1.33481i) q^{7} +(-9.73548 + 16.8623i) q^{9} +O(q^{10})\) \(q+(-3.40848 - 5.90366i) q^{3} +12.7779i q^{5} +(2.31195 + 1.33481i) q^{7} +(-9.73548 + 16.8623i) q^{9} +(34.8449 - 20.1177i) q^{11} +(25.0545 - 39.6140i) q^{13} +(75.4364 - 43.5532i) q^{15} +(-16.4208 + 28.4416i) q^{17} +(-15.5366 - 8.97007i) q^{19} -18.1987i q^{21} +(-68.7952 - 119.157i) q^{23} -38.2746 q^{25} -51.3252 q^{27} +(-51.2989 - 88.8524i) q^{29} -276.308i q^{31} +(-237.536 - 137.142i) q^{33} +(-17.0560 + 29.5419i) q^{35} +(350.213 - 202.195i) q^{37} +(-319.266 - 12.8900i) q^{39} +(8.94701 - 5.16556i) q^{41} +(-102.049 + 176.754i) q^{43} +(-215.465 - 124.399i) q^{45} +160.135i q^{47} +(-167.937 - 290.875i) q^{49} +223.880 q^{51} +547.517 q^{53} +(257.062 + 445.244i) q^{55} +122.297i q^{57} +(356.890 + 206.051i) q^{59} +(302.095 - 523.245i) q^{61} +(-45.0159 + 25.9900i) q^{63} +(506.184 + 320.144i) q^{65} +(584.393 - 337.399i) q^{67} +(-468.974 + 812.287i) q^{69} +(-624.030 - 360.284i) q^{71} +562.708i q^{73} +(130.458 + 225.960i) q^{75} +107.413 q^{77} -587.720 q^{79} +(437.799 + 758.290i) q^{81} -583.213i q^{83} +(-363.424 - 209.823i) q^{85} +(-349.703 + 605.703i) q^{87} +(-425.924 + 245.907i) q^{89} +(110.802 - 58.1428i) q^{91} +(-1631.23 + 941.792i) q^{93} +(114.619 - 198.525i) q^{95} +(-1478.34 - 853.519i) q^{97} +783.422i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{3} + 54 q^{7} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{3} + 54 q^{7} - 72 q^{9} + 42 q^{11} + 24 q^{13} - 70 q^{17} + 102 q^{19} - 90 q^{23} - 628 q^{25} - 708 q^{27} + 170 q^{29} - 678 q^{33} + 544 q^{35} + 582 q^{37} - 1162 q^{39} + 438 q^{41} - 270 q^{43} + 540 q^{45} + 92 q^{49} + 444 q^{51} - 592 q^{53} + 288 q^{55} - 90 q^{59} + 746 q^{61} + 1068 q^{63} - 1412 q^{65} - 846 q^{67} + 682 q^{69} + 1038 q^{71} - 722 q^{75} + 2812 q^{77} + 1008 q^{79} + 694 q^{81} - 180 q^{85} + 338 q^{87} + 2466 q^{89} - 690 q^{91} - 4764 q^{93} + 2592 q^{95} - 846 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −3.40848 5.90366i −0.655962 1.13616i −0.981652 0.190683i \(-0.938930\pi\)
0.325689 0.945477i \(-0.394404\pi\)
\(4\) 0 0
\(5\) 12.7779i 1.14289i 0.820641 + 0.571445i \(0.193617\pi\)
−0.820641 + 0.571445i \(0.806383\pi\)
\(6\) 0 0
\(7\) 2.31195 + 1.33481i 0.124834 + 0.0720728i 0.561117 0.827737i \(-0.310372\pi\)
−0.436283 + 0.899810i \(0.643705\pi\)
\(8\) 0 0
\(9\) −9.73548 + 16.8623i −0.360573 + 0.624531i
\(10\) 0 0
\(11\) 34.8449 20.1177i 0.955103 0.551429i 0.0604407 0.998172i \(-0.480749\pi\)
0.894662 + 0.446743i \(0.147416\pi\)
\(12\) 0 0
\(13\) 25.0545 39.6140i 0.534529 0.845150i
\(14\) 0 0
\(15\) 75.4364 43.5532i 1.29851 0.749693i
\(16\) 0 0
\(17\) −16.4208 + 28.4416i −0.234272 + 0.405771i −0.959061 0.283200i \(-0.908604\pi\)
0.724789 + 0.688971i \(0.241937\pi\)
\(18\) 0 0
\(19\) −15.5366 8.97007i −0.187597 0.108309i 0.403260 0.915085i \(-0.367877\pi\)
−0.590857 + 0.806776i \(0.701210\pi\)
\(20\) 0 0
\(21\) 18.1987i 0.189108i
\(22\) 0 0
\(23\) −68.7952 119.157i −0.623686 1.08026i −0.988793 0.149291i \(-0.952301\pi\)
0.365107 0.930966i \(-0.381032\pi\)
\(24\) 0 0
\(25\) −38.2746 −0.306196
\(26\) 0 0
\(27\) −51.3252 −0.365835
\(28\) 0 0
\(29\) −51.2989 88.8524i −0.328482 0.568947i 0.653729 0.756729i \(-0.273204\pi\)
−0.982211 + 0.187781i \(0.939870\pi\)
\(30\) 0 0
\(31\) 276.308i 1.60085i −0.599431 0.800427i \(-0.704606\pi\)
0.599431 0.800427i \(-0.295394\pi\)
\(32\) 0 0
\(33\) −237.536 137.142i −1.25302 0.723433i
\(34\) 0 0
\(35\) −17.0560 + 29.5419i −0.0823712 + 0.142671i
\(36\) 0 0
\(37\) 350.213 202.195i 1.55607 0.898398i 0.558444 0.829542i \(-0.311399\pi\)
0.997627 0.0688555i \(-0.0219347\pi\)
\(38\) 0 0
\(39\) −319.266 12.8900i −1.31086 0.0529243i
\(40\) 0 0
\(41\) 8.94701 5.16556i 0.0340802 0.0196762i −0.482863 0.875696i \(-0.660403\pi\)
0.516943 + 0.856020i \(0.327070\pi\)
\(42\) 0 0
\(43\) −102.049 + 176.754i −0.361915 + 0.626855i −0.988276 0.152678i \(-0.951210\pi\)
0.626361 + 0.779533i \(0.284543\pi\)
\(44\) 0 0
\(45\) −215.465 124.399i −0.713770 0.412095i
\(46\) 0 0
\(47\) 160.135i 0.496982i 0.968634 + 0.248491i \(0.0799346\pi\)
−0.968634 + 0.248491i \(0.920065\pi\)
\(48\) 0 0
\(49\) −167.937 290.875i −0.489611 0.848031i
\(50\) 0 0
\(51\) 223.880 0.614694
\(52\) 0 0
\(53\) 547.517 1.41900 0.709502 0.704703i \(-0.248920\pi\)
0.709502 + 0.704703i \(0.248920\pi\)
\(54\) 0 0
\(55\) 257.062 + 445.244i 0.630222 + 1.09158i
\(56\) 0 0
\(57\) 122.297i 0.284187i
\(58\) 0 0
\(59\) 356.890 + 206.051i 0.787511 + 0.454670i 0.839085 0.544000i \(-0.183091\pi\)
−0.0515747 + 0.998669i \(0.516424\pi\)
\(60\) 0 0
\(61\) 302.095 523.245i 0.634088 1.09827i −0.352620 0.935767i \(-0.614709\pi\)
0.986708 0.162506i \(-0.0519575\pi\)
\(62\) 0 0
\(63\) −45.0159 + 25.9900i −0.0900234 + 0.0519750i
\(64\) 0 0
\(65\) 506.184 + 320.144i 0.965913 + 0.610908i
\(66\) 0 0
\(67\) 584.393 337.399i 1.06560 0.615222i 0.138621 0.990345i \(-0.455733\pi\)
0.926975 + 0.375123i \(0.122400\pi\)
\(68\) 0 0
\(69\) −468.974 + 812.287i −0.818230 + 1.41722i
\(70\) 0 0
\(71\) −624.030 360.284i −1.04308 0.602223i −0.122376 0.992484i \(-0.539051\pi\)
−0.920704 + 0.390261i \(0.872385\pi\)
\(72\) 0 0
\(73\) 562.708i 0.902191i 0.892476 + 0.451096i \(0.148967\pi\)
−0.892476 + 0.451096i \(0.851033\pi\)
\(74\) 0 0
\(75\) 130.458 + 225.960i 0.200853 + 0.347888i
\(76\) 0 0
\(77\) 107.413 0.158972
\(78\) 0 0
\(79\) −587.720 −0.837008 −0.418504 0.908215i \(-0.637445\pi\)
−0.418504 + 0.908215i \(0.637445\pi\)
\(80\) 0 0
\(81\) 437.799 + 758.290i 0.600547 + 1.04018i
\(82\) 0 0
\(83\) 583.213i 0.771276i −0.922650 0.385638i \(-0.873981\pi\)
0.922650 0.385638i \(-0.126019\pi\)
\(84\) 0 0
\(85\) −363.424 209.823i −0.463751 0.267747i
\(86\) 0 0
\(87\) −349.703 + 605.703i −0.430944 + 0.746416i
\(88\) 0 0
\(89\) −425.924 + 245.907i −0.507279 + 0.292878i −0.731714 0.681611i \(-0.761280\pi\)
0.224436 + 0.974489i \(0.427946\pi\)
\(90\) 0 0
\(91\) 110.802 58.1428i 0.127640 0.0669782i
\(92\) 0 0
\(93\) −1631.23 + 941.792i −1.81883 + 1.05010i
\(94\) 0 0
\(95\) 114.619 198.525i 0.123785 0.214403i
\(96\) 0 0
\(97\) −1478.34 853.519i −1.54745 0.893420i −0.998336 0.0576723i \(-0.981632\pi\)
−0.549113 0.835748i \(-0.685035\pi\)
\(98\) 0 0
\(99\) 783.422i 0.795322i
\(100\) 0 0
\(101\) 607.983 + 1053.06i 0.598976 + 1.03746i 0.992973 + 0.118345i \(0.0377588\pi\)
−0.393997 + 0.919112i \(0.628908\pi\)
\(102\) 0 0
\(103\) 778.221 0.744469 0.372235 0.928139i \(-0.378592\pi\)
0.372235 + 0.928139i \(0.378592\pi\)
\(104\) 0 0
\(105\) 232.540 0.216130
\(106\) 0 0
\(107\) 982.647 + 1702.00i 0.887814 + 1.53774i 0.842454 + 0.538768i \(0.181110\pi\)
0.0453600 + 0.998971i \(0.485557\pi\)
\(108\) 0 0
\(109\) 434.187i 0.381537i −0.981635 0.190769i \(-0.938902\pi\)
0.981635 0.190769i \(-0.0610980\pi\)
\(110\) 0 0
\(111\) −2387.39 1378.36i −2.04145 1.17863i
\(112\) 0 0
\(113\) 690.378 1195.77i 0.574737 0.995473i −0.421333 0.906906i \(-0.638438\pi\)
0.996070 0.0885675i \(-0.0282289\pi\)
\(114\) 0 0
\(115\) 1522.57 879.057i 1.23461 0.712805i
\(116\) 0 0
\(117\) 424.067 + 808.140i 0.335086 + 0.638569i
\(118\) 0 0
\(119\) −75.9281 + 43.8371i −0.0584901 + 0.0337693i
\(120\) 0 0
\(121\) 143.945 249.320i 0.108148 0.187318i
\(122\) 0 0
\(123\) −60.9914 35.2134i −0.0447106 0.0258137i
\(124\) 0 0
\(125\) 1108.17i 0.792941i
\(126\) 0 0
\(127\) 296.754 + 513.993i 0.207344 + 0.359130i 0.950877 0.309569i \(-0.100185\pi\)
−0.743533 + 0.668699i \(0.766851\pi\)
\(128\) 0 0
\(129\) 1391.33 0.949610
\(130\) 0 0
\(131\) −134.098 −0.0894366 −0.0447183 0.999000i \(-0.514239\pi\)
−0.0447183 + 0.999000i \(0.514239\pi\)
\(132\) 0 0
\(133\) −23.9466 41.4768i −0.0156123 0.0270413i
\(134\) 0 0
\(135\) 655.828i 0.418109i
\(136\) 0 0
\(137\) 1744.68 + 1007.29i 1.08801 + 0.628166i 0.933047 0.359754i \(-0.117139\pi\)
0.154968 + 0.987920i \(0.450473\pi\)
\(138\) 0 0
\(139\) −1321.63 + 2289.13i −0.806471 + 1.39685i 0.108823 + 0.994061i \(0.465292\pi\)
−0.915294 + 0.402787i \(0.868042\pi\)
\(140\) 0 0
\(141\) 945.385 545.819i 0.564651 0.326002i
\(142\) 0 0
\(143\) 76.0799 1884.39i 0.0444904 1.10196i
\(144\) 0 0
\(145\) 1135.35 655.492i 0.650244 0.375419i
\(146\) 0 0
\(147\) −1144.82 + 1982.88i −0.642333 + 1.11255i
\(148\) 0 0
\(149\) −2444.08 1411.09i −1.34380 0.775844i −0.356439 0.934319i \(-0.616009\pi\)
−0.987363 + 0.158474i \(0.949342\pi\)
\(150\) 0 0
\(151\) 3193.28i 1.72096i 0.509483 + 0.860481i \(0.329837\pi\)
−0.509483 + 0.860481i \(0.670163\pi\)
\(152\) 0 0
\(153\) −319.728 553.785i −0.168944 0.292620i
\(154\) 0 0
\(155\) 3530.64 1.82960
\(156\) 0 0
\(157\) −2027.72 −1.03076 −0.515381 0.856961i \(-0.672350\pi\)
−0.515381 + 0.856961i \(0.672350\pi\)
\(158\) 0 0
\(159\) −1866.20 3232.35i −0.930813 1.61222i
\(160\) 0 0
\(161\) 367.313i 0.179803i
\(162\) 0 0
\(163\) 517.804 + 298.954i 0.248819 + 0.143656i 0.619223 0.785215i \(-0.287448\pi\)
−0.370404 + 0.928871i \(0.620781\pi\)
\(164\) 0 0
\(165\) 1752.38 3035.21i 0.826804 1.43207i
\(166\) 0 0
\(167\) 899.026 519.053i 0.416579 0.240512i −0.277033 0.960860i \(-0.589351\pi\)
0.693613 + 0.720348i \(0.256018\pi\)
\(168\) 0 0
\(169\) −941.539 1985.02i −0.428557 0.903515i
\(170\) 0 0
\(171\) 302.513 174.656i 0.135285 0.0781068i
\(172\) 0 0
\(173\) 553.114 958.022i 0.243078 0.421023i −0.718512 0.695515i \(-0.755176\pi\)
0.961589 + 0.274492i \(0.0885096\pi\)
\(174\) 0 0
\(175\) −88.4890 51.0891i −0.0382236 0.0220684i
\(176\) 0 0
\(177\) 2809.28i 1.19298i
\(178\) 0 0
\(179\) −771.535 1336.34i −0.322163 0.558003i 0.658771 0.752344i \(-0.271077\pi\)
−0.980934 + 0.194341i \(0.937743\pi\)
\(180\) 0 0
\(181\) −813.346 −0.334009 −0.167004 0.985956i \(-0.553409\pi\)
−0.167004 + 0.985956i \(0.553409\pi\)
\(182\) 0 0
\(183\) −4118.74 −1.66375
\(184\) 0 0
\(185\) 2583.63 + 4474.98i 1.02677 + 1.77842i
\(186\) 0 0
\(187\) 1321.39i 0.516737i
\(188\) 0 0
\(189\) −118.661 68.5092i −0.0456685 0.0263667i
\(190\) 0 0
\(191\) 1119.72 1939.41i 0.424188 0.734715i −0.572157 0.820144i \(-0.693893\pi\)
0.996344 + 0.0854299i \(0.0272264\pi\)
\(192\) 0 0
\(193\) −3123.85 + 1803.56i −1.16508 + 0.672657i −0.952515 0.304491i \(-0.901514\pi\)
−0.212561 + 0.977148i \(0.568180\pi\)
\(194\) 0 0
\(195\) 164.707 4079.54i 0.0604866 1.49816i
\(196\) 0 0
\(197\) 1838.32 1061.35i 0.664847 0.383850i −0.129274 0.991609i \(-0.541265\pi\)
0.794121 + 0.607759i \(0.207931\pi\)
\(198\) 0 0
\(199\) 296.559 513.656i 0.105641 0.182975i −0.808359 0.588690i \(-0.799644\pi\)
0.914000 + 0.405714i \(0.132977\pi\)
\(200\) 0 0
\(201\) −3983.78 2300.04i −1.39798 0.807125i
\(202\) 0 0
\(203\) 273.897i 0.0946984i
\(204\) 0 0
\(205\) 66.0049 + 114.324i 0.0224877 + 0.0389499i
\(206\) 0 0
\(207\) 2679.02 0.899538
\(208\) 0 0
\(209\) −721.829 −0.238899
\(210\) 0 0
\(211\) −2502.07 4333.72i −0.816350 1.41396i −0.908354 0.418201i \(-0.862661\pi\)
0.0920042 0.995759i \(-0.470673\pi\)
\(212\) 0 0
\(213\) 4912.08i 1.58014i
\(214\) 0 0
\(215\) −2258.54 1303.97i −0.716426 0.413628i
\(216\) 0 0
\(217\) 368.818 638.812i 0.115378 0.199840i
\(218\) 0 0
\(219\) 3322.04 1917.98i 1.02503 0.591804i
\(220\) 0 0
\(221\) 715.271 + 1363.08i 0.217712 + 0.414891i
\(222\) 0 0
\(223\) 408.594 235.902i 0.122697 0.0708392i −0.437395 0.899269i \(-0.644099\pi\)
0.560092 + 0.828430i \(0.310766\pi\)
\(224\) 0 0
\(225\) 372.621 645.399i 0.110406 0.191229i
\(226\) 0 0
\(227\) 2505.80 + 1446.72i 0.732668 + 0.423006i 0.819397 0.573226i \(-0.194308\pi\)
−0.0867293 + 0.996232i \(0.527642\pi\)
\(228\) 0 0
\(229\) 2444.04i 0.705269i −0.935761 0.352635i \(-0.885286\pi\)
0.935761 0.352635i \(-0.114714\pi\)
\(230\) 0 0
\(231\) −366.115 634.130i −0.104280 0.180618i
\(232\) 0 0
\(233\) −2894.86 −0.813944 −0.406972 0.913441i \(-0.633415\pi\)
−0.406972 + 0.913441i \(0.633415\pi\)
\(234\) 0 0
\(235\) −2046.19 −0.567996
\(236\) 0 0
\(237\) 2003.23 + 3469.70i 0.549046 + 0.950975i
\(238\) 0 0
\(239\) 2819.27i 0.763027i 0.924363 + 0.381513i \(0.124597\pi\)
−0.924363 + 0.381513i \(0.875403\pi\)
\(240\) 0 0
\(241\) −39.0443 22.5423i −0.0104360 0.00602521i 0.494773 0.869022i \(-0.335251\pi\)
−0.505209 + 0.862997i \(0.668585\pi\)
\(242\) 0 0
\(243\) 2291.57 3969.11i 0.604955 1.04781i
\(244\) 0 0
\(245\) 3716.77 2145.88i 0.969206 0.559571i
\(246\) 0 0
\(247\) −744.603 + 390.727i −0.191814 + 0.100653i
\(248\) 0 0
\(249\) −3443.09 + 1987.87i −0.876293 + 0.505928i
\(250\) 0 0
\(251\) −2951.26 + 5111.74i −0.742160 + 1.28546i 0.209351 + 0.977841i \(0.432865\pi\)
−0.951510 + 0.307617i \(0.900468\pi\)
\(252\) 0 0
\(253\) −4794.32 2768.00i −1.19137 0.687838i
\(254\) 0 0
\(255\) 2860.71i 0.702528i
\(256\) 0 0
\(257\) 2570.84 + 4452.82i 0.623986 + 1.08078i 0.988736 + 0.149670i \(0.0478210\pi\)
−0.364750 + 0.931105i \(0.618846\pi\)
\(258\) 0 0
\(259\) 1079.57 0.259000
\(260\) 0 0
\(261\) 1997.68 0.473767
\(262\) 0 0
\(263\) 3073.40 + 5323.28i 0.720585 + 1.24809i 0.960766 + 0.277362i \(0.0894600\pi\)
−0.240181 + 0.970728i \(0.577207\pi\)
\(264\) 0 0
\(265\) 6996.11i 1.62177i
\(266\) 0 0
\(267\) 2903.50 + 1676.34i 0.665512 + 0.384233i
\(268\) 0 0
\(269\) 468.880 812.125i 0.106276 0.184075i −0.807983 0.589206i \(-0.799441\pi\)
0.914259 + 0.405131i \(0.132774\pi\)
\(270\) 0 0
\(271\) −1526.89 + 881.548i −0.342257 + 0.197602i −0.661270 0.750148i \(-0.729982\pi\)
0.319012 + 0.947751i \(0.396649\pi\)
\(272\) 0 0
\(273\) −720.921 455.959i −0.159825 0.101084i
\(274\) 0 0
\(275\) −1333.67 + 769.997i −0.292449 + 0.168846i
\(276\) 0 0
\(277\) 3298.97 5713.98i 0.715581 1.23942i −0.247154 0.968976i \(-0.579495\pi\)
0.962735 0.270446i \(-0.0871713\pi\)
\(278\) 0 0
\(279\) 4659.21 + 2689.99i 0.999783 + 0.577225i
\(280\) 0 0
\(281\) 4163.19i 0.883826i −0.897058 0.441913i \(-0.854300\pi\)
0.897058 0.441913i \(-0.145700\pi\)
\(282\) 0 0
\(283\) −2958.33 5123.98i −0.621394 1.07629i −0.989226 0.146393i \(-0.953233\pi\)
0.367833 0.929892i \(-0.380100\pi\)
\(284\) 0 0
\(285\) −1562.70 −0.324794
\(286\) 0 0
\(287\) 27.5801 0.00567247
\(288\) 0 0
\(289\) 1917.22 + 3320.72i 0.390233 + 0.675904i
\(290\) 0 0
\(291\) 11636.8i 2.34420i
\(292\) 0 0
\(293\) 5937.45 + 3427.99i 1.18386 + 0.683499i 0.956903 0.290406i \(-0.0937905\pi\)
0.226952 + 0.973906i \(0.427124\pi\)
\(294\) 0 0
\(295\) −2632.89 + 4560.30i −0.519637 + 0.900038i
\(296\) 0 0
\(297\) −1788.42 + 1032.55i −0.349410 + 0.201732i
\(298\) 0 0
\(299\) −6443.91 260.165i −1.24636 0.0503202i
\(300\) 0 0
\(301\) −471.865 + 272.431i −0.0903583 + 0.0521684i
\(302\) 0 0
\(303\) 4144.60 7178.65i 0.785811 1.36106i
\(304\) 0 0
\(305\) 6685.96 + 3860.14i 1.25520 + 0.724692i
\(306\) 0 0
\(307\) 2846.57i 0.529194i 0.964359 + 0.264597i \(0.0852389\pi\)
−0.964359 + 0.264597i \(0.914761\pi\)
\(308\) 0 0
\(309\) −2652.55 4594.35i −0.488344 0.845836i
\(310\) 0 0
\(311\) 6018.53 1.09736 0.548681 0.836032i \(-0.315130\pi\)
0.548681 + 0.836032i \(0.315130\pi\)
\(312\) 0 0
\(313\) −2637.18 −0.476238 −0.238119 0.971236i \(-0.576531\pi\)
−0.238119 + 0.971236i \(0.576531\pi\)
\(314\) 0 0
\(315\) −332.097 575.209i −0.0594017 0.102887i
\(316\) 0 0
\(317\) 9016.09i 1.59746i 0.601692 + 0.798728i \(0.294494\pi\)
−0.601692 + 0.798728i \(0.705506\pi\)
\(318\) 0 0
\(319\) −3575.01 2064.04i −0.627468 0.362269i
\(320\) 0 0
\(321\) 6698.67 11602.4i 1.16475 2.01740i
\(322\) 0 0
\(323\) 510.247 294.591i 0.0878974 0.0507476i
\(324\) 0 0
\(325\) −958.952 + 1516.21i −0.163671 + 0.258782i
\(326\) 0 0
\(327\) −2563.29 + 1479.92i −0.433487 + 0.250274i
\(328\) 0 0
\(329\) −213.750 + 370.226i −0.0358189 + 0.0620401i
\(330\) 0 0
\(331\) −4457.08 2573.30i −0.740131 0.427315i 0.0819859 0.996633i \(-0.473874\pi\)
−0.822117 + 0.569319i \(0.807207\pi\)
\(332\) 0 0
\(333\) 7873.87i 1.29575i
\(334\) 0 0
\(335\) 4311.25 + 7467.31i 0.703131 + 1.21786i
\(336\) 0 0
\(337\) 9709.21 1.56942 0.784710 0.619863i \(-0.212812\pi\)
0.784710 + 0.619863i \(0.212812\pi\)
\(338\) 0 0
\(339\) −9412.55 −1.50802
\(340\) 0 0
\(341\) −5558.69 9627.94i −0.882757 1.52898i
\(342\) 0 0
\(343\) 1812.33i 0.285296i
\(344\) 0 0
\(345\) −10379.3 5992.50i −1.61972 0.935146i
\(346\) 0 0
\(347\) −3406.91 + 5900.94i −0.527068 + 0.912908i 0.472435 + 0.881366i \(0.343375\pi\)
−0.999502 + 0.0315424i \(0.989958\pi\)
\(348\) 0 0
\(349\) −2110.42 + 1218.45i −0.323691 + 0.186883i −0.653037 0.757326i \(-0.726505\pi\)
0.329345 + 0.944210i \(0.393172\pi\)
\(350\) 0 0
\(351\) −1285.93 + 2033.20i −0.195549 + 0.309185i
\(352\) 0 0
\(353\) −8711.72 + 5029.71i −1.31354 + 0.758370i −0.982680 0.185311i \(-0.940671\pi\)
−0.330856 + 0.943681i \(0.607337\pi\)
\(354\) 0 0
\(355\) 4603.67 7973.79i 0.688274 1.19213i
\(356\) 0 0
\(357\) 517.599 + 298.836i 0.0767346 + 0.0443027i
\(358\) 0 0
\(359\) 2429.18i 0.357123i −0.983929 0.178562i \(-0.942856\pi\)
0.983929 0.178562i \(-0.0571444\pi\)
\(360\) 0 0
\(361\) −3268.58 5661.34i −0.476538 0.825388i
\(362\) 0 0
\(363\) −1962.53 −0.283764
\(364\) 0 0
\(365\) −7190.22 −1.03110
\(366\) 0 0
\(367\) 1312.28 + 2272.94i 0.186650 + 0.323288i 0.944131 0.329569i \(-0.106904\pi\)
−0.757481 + 0.652857i \(0.773570\pi\)
\(368\) 0 0
\(369\) 201.157i 0.0283788i
\(370\) 0 0
\(371\) 1265.83 + 730.829i 0.177140 + 0.102272i
\(372\) 0 0
\(373\) −3787.69 + 6560.46i −0.525788 + 0.910692i 0.473761 + 0.880654i \(0.342896\pi\)
−0.999549 + 0.0300380i \(0.990437\pi\)
\(374\) 0 0
\(375\) 6542.25 3777.17i 0.900908 0.520139i
\(376\) 0 0
\(377\) −4805.07 193.999i −0.656429 0.0265026i
\(378\) 0 0
\(379\) −587.129 + 338.979i −0.0795746 + 0.0459424i −0.539259 0.842140i \(-0.681296\pi\)
0.459685 + 0.888082i \(0.347962\pi\)
\(380\) 0 0
\(381\) 2022.96 3503.87i 0.272020 0.471152i
\(382\) 0 0
\(383\) 5093.76 + 2940.88i 0.679579 + 0.392355i 0.799697 0.600404i \(-0.204994\pi\)
−0.120117 + 0.992760i \(0.538327\pi\)
\(384\) 0 0
\(385\) 1372.51i 0.181688i
\(386\) 0 0
\(387\) −1986.99 3441.57i −0.260993 0.452054i
\(388\) 0 0
\(389\) 6623.27 0.863273 0.431637 0.902048i \(-0.357936\pi\)
0.431637 + 0.902048i \(0.357936\pi\)
\(390\) 0 0
\(391\) 4518.68 0.584449
\(392\) 0 0
\(393\) 457.070 + 791.669i 0.0586670 + 0.101614i
\(394\) 0 0
\(395\) 7509.82i 0.956608i
\(396\) 0 0
\(397\) 4723.28 + 2726.99i 0.597115 + 0.344744i 0.767906 0.640563i \(-0.221299\pi\)
−0.170791 + 0.985307i \(0.554632\pi\)
\(398\) 0 0
\(399\) −163.243 + 282.745i −0.0204822 + 0.0354761i
\(400\) 0 0
\(401\) 8578.70 4952.92i 1.06833 0.616800i 0.140604 0.990066i \(-0.455095\pi\)
0.927724 + 0.373266i \(0.121762\pi\)
\(402\) 0 0
\(403\) −10945.7 6922.78i −1.35296 0.855703i
\(404\) 0 0
\(405\) −9689.35 + 5594.15i −1.18881 + 0.686359i
\(406\) 0 0
\(407\) 8135.42 14091.0i 0.990805 1.71613i
\(408\) 0 0
\(409\) 12274.0 + 7086.42i 1.48389 + 0.856726i 0.999832 0.0183077i \(-0.00582783\pi\)
0.484061 + 0.875034i \(0.339161\pi\)
\(410\) 0 0
\(411\) 13733.3i 1.64821i
\(412\) 0 0
\(413\) 550.075 + 952.759i 0.0655386 + 0.113516i
\(414\) 0 0
\(415\) 7452.23 0.881483
\(416\) 0 0
\(417\) 18019.0 2.11606
\(418\) 0 0
\(419\) 5639.55 + 9767.99i 0.657542 + 1.13890i 0.981250 + 0.192740i \(0.0617372\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(420\) 0 0
\(421\) 2422.46i 0.280436i −0.990121 0.140218i \(-0.955220\pi\)
0.990121 0.140218i \(-0.0447803\pi\)
\(422\) 0 0
\(423\) −2700.26 1559.00i −0.310381 0.179198i
\(424\) 0 0
\(425\) 628.498 1088.59i 0.0717332 0.124246i
\(426\) 0 0
\(427\) 1396.86 806.478i 0.158311 0.0914009i
\(428\) 0 0
\(429\) −11384.1 + 5973.75i −1.28119 + 0.672296i
\(430\) 0 0
\(431\) −1254.01 + 724.003i −0.140147 + 0.0809141i −0.568434 0.822729i \(-0.692451\pi\)
0.428287 + 0.903643i \(0.359117\pi\)
\(432\) 0 0
\(433\) −2270.03 + 3931.81i −0.251942 + 0.436376i −0.964060 0.265683i \(-0.914402\pi\)
0.712119 + 0.702059i \(0.247736\pi\)
\(434\) 0 0
\(435\) −7739.61 4468.47i −0.853071 0.492521i
\(436\) 0 0
\(437\) 2468.39i 0.270204i
\(438\) 0 0
\(439\) −5058.37 8761.35i −0.549938 0.952520i −0.998278 0.0586574i \(-0.981318\pi\)
0.448340 0.893863i \(-0.352015\pi\)
\(440\) 0 0
\(441\) 6539.77 0.706163
\(442\) 0 0
\(443\) 18380.3 1.97127 0.985636 0.168884i \(-0.0540164\pi\)
0.985636 + 0.168884i \(0.0540164\pi\)
\(444\) 0 0
\(445\) −3142.17 5442.41i −0.334727 0.579764i
\(446\) 0 0
\(447\) 19238.7i 2.03570i
\(448\) 0 0
\(449\) −3868.31 2233.37i −0.406586 0.234742i 0.282736 0.959198i \(-0.408758\pi\)
−0.689322 + 0.724455i \(0.742091\pi\)
\(450\) 0 0
\(451\) 207.838 359.987i 0.0217001 0.0375856i
\(452\) 0 0
\(453\) 18852.0 10884.2i 1.95529 1.12889i
\(454\) 0 0
\(455\) 742.942 + 1415.82i 0.0765487 + 0.145878i
\(456\) 0 0
\(457\) −7125.63 + 4113.98i −0.729371 + 0.421103i −0.818192 0.574945i \(-0.805024\pi\)
0.0888207 + 0.996048i \(0.471690\pi\)
\(458\) 0 0
\(459\) 842.800 1459.77i 0.0857048 0.148445i
\(460\) 0 0
\(461\) −13839.2 7990.04i −1.39816 0.807231i −0.403964 0.914775i \(-0.632368\pi\)
−0.994200 + 0.107544i \(0.965701\pi\)
\(462\) 0 0
\(463\) 8317.95i 0.834920i −0.908695 0.417460i \(-0.862920\pi\)
0.908695 0.417460i \(-0.137080\pi\)
\(464\) 0 0
\(465\) −12034.1 20843.7i −1.20015 2.07872i
\(466\) 0 0
\(467\) −7477.40 −0.740927 −0.370464 0.928847i \(-0.620801\pi\)
−0.370464 + 0.928847i \(0.620801\pi\)
\(468\) 0 0
\(469\) 1801.45 0.177363
\(470\) 0 0
\(471\) 6911.44 + 11971.0i 0.676141 + 1.17111i
\(472\) 0 0
\(473\) 8211.97i 0.798281i
\(474\) 0 0
\(475\) 594.657 + 343.325i 0.0574416 + 0.0331639i
\(476\) 0 0
\(477\) −5330.34 + 9232.42i −0.511655 + 0.886212i
\(478\) 0 0
\(479\) −3124.69 + 1804.04i −0.298060 + 0.172085i −0.641571 0.767064i \(-0.721717\pi\)
0.343511 + 0.939149i \(0.388384\pi\)
\(480\) 0 0
\(481\) 764.650 18939.2i 0.0724845 1.79533i
\(482\) 0 0
\(483\) −2168.49 + 1251.98i −0.204285 + 0.117944i
\(484\) 0 0
\(485\) 10906.2 18890.1i 1.02108 1.76856i
\(486\) 0 0
\(487\) −6596.81 3808.67i −0.613819 0.354389i 0.160640 0.987013i \(-0.448644\pi\)
−0.774459 + 0.632624i \(0.781978\pi\)
\(488\) 0 0
\(489\) 4075.92i 0.376931i
\(490\) 0 0
\(491\) −4797.86 8310.14i −0.440987 0.763811i 0.556776 0.830663i \(-0.312038\pi\)
−0.997763 + 0.0668511i \(0.978705\pi\)
\(492\) 0 0
\(493\) 3369.47 0.307816
\(494\) 0 0
\(495\) −10010.5 −0.908965
\(496\) 0 0
\(497\) −961.819 1665.92i −0.0868078 0.150355i
\(498\) 0 0
\(499\) 5761.93i 0.516913i 0.966023 + 0.258457i \(0.0832139\pi\)
−0.966023 + 0.258457i \(0.916786\pi\)
\(500\) 0 0
\(501\) −6128.63 3538.36i −0.546521 0.315534i
\(502\) 0 0
\(503\) 3958.66 6856.61i 0.350911 0.607795i −0.635499 0.772102i \(-0.719205\pi\)
0.986409 + 0.164307i \(0.0525387\pi\)
\(504\) 0 0
\(505\) −13455.9 + 7768.74i −1.18570 + 0.684563i
\(506\) 0 0
\(507\) −8509.68 + 12324.4i −0.745420 + 1.07958i
\(508\) 0 0
\(509\) 3408.98 1968.18i 0.296858 0.171391i −0.344173 0.938906i \(-0.611841\pi\)
0.641030 + 0.767516i \(0.278507\pi\)
\(510\) 0 0
\(511\) −751.106 + 1300.95i −0.0650234 + 0.112624i
\(512\) 0 0
\(513\) 797.420 + 460.391i 0.0686295 + 0.0396233i
\(514\) 0 0
\(515\) 9944.02i 0.850846i
\(516\) 0 0
\(517\) 3221.56 + 5579.90i 0.274050 + 0.474669i
\(518\) 0 0
\(519\) −7541.11 −0.637800
\(520\) 0 0
\(521\) 2588.98 0.217707 0.108853 0.994058i \(-0.465282\pi\)
0.108853 + 0.994058i \(0.465282\pi\)
\(522\) 0 0
\(523\) −6222.45 10777.6i −0.520246 0.901093i −0.999723 0.0235385i \(-0.992507\pi\)
0.479476 0.877555i \(-0.340827\pi\)
\(524\) 0 0
\(525\) 696.545i 0.0579042i
\(526\) 0 0
\(527\) 7858.66 + 4537.20i 0.649580 + 0.375035i
\(528\) 0 0
\(529\) −3382.05 + 5857.89i −0.277969 + 0.481457i
\(530\) 0 0
\(531\) −6948.99 + 4012.00i −0.567911 + 0.327883i
\(532\) 0 0
\(533\) 19.5348 483.847i 0.00158751 0.0393204i
\(534\) 0 0
\(535\) −21747.9 + 12556.2i −1.75747 + 1.01467i
\(536\) 0 0
\(537\) −5259.52 + 9109.76i −0.422654 + 0.732058i
\(538\) 0 0
\(539\) −11703.5 6757.00i −0.935258 0.539971i
\(540\) 0 0
\(541\) 2221.88i 0.176573i −0.996095 0.0882866i \(-0.971861\pi\)
0.996095 0.0882866i \(-0.0281391\pi\)
\(542\) 0 0
\(543\) 2772.27 + 4801.72i 0.219097 + 0.379487i
\(544\) 0 0
\(545\) 5547.99 0.436055
\(546\) 0 0
\(547\) 11535.2 0.901661 0.450830 0.892610i \(-0.351128\pi\)
0.450830 + 0.892610i \(0.351128\pi\)
\(548\) 0 0
\(549\) 5882.09 + 10188.1i 0.457270 + 0.792015i
\(550\) 0 0
\(551\) 1840.62i 0.142310i
\(552\) 0 0
\(553\) −1358.78 784.492i −0.104487 0.0603255i
\(554\) 0 0
\(555\) 17612.5 30505.8i 1.34704 2.33315i
\(556\) 0 0
\(557\) 11299.7 6523.89i 0.859577 0.496277i −0.00429384 0.999991i \(-0.501367\pi\)
0.863870 + 0.503714i \(0.168033\pi\)
\(558\) 0 0
\(559\) 4445.15 + 8471.06i 0.336332 + 0.640944i
\(560\) 0 0
\(561\) 7801.06 4503.95i 0.587096 0.338960i
\(562\) 0 0
\(563\) −4693.55 + 8129.46i −0.351349 + 0.608554i −0.986486 0.163845i \(-0.947610\pi\)
0.635137 + 0.772399i \(0.280944\pi\)
\(564\) 0 0
\(565\) 15279.4 + 8821.57i 1.13772 + 0.656861i
\(566\) 0 0
\(567\) 2337.51i 0.173132i
\(568\) 0 0
\(569\) 505.020 + 874.719i 0.0372083 + 0.0644467i 0.884030 0.467430i \(-0.154820\pi\)
−0.846822 + 0.531877i \(0.821487\pi\)
\(570\) 0 0
\(571\) 17206.4 1.26106 0.630529 0.776166i \(-0.282838\pi\)
0.630529 + 0.776166i \(0.282838\pi\)
\(572\) 0 0
\(573\) −15266.1 −1.11300
\(574\) 0 0
\(575\) 2633.11 + 4560.67i 0.190971 + 0.330771i
\(576\) 0 0
\(577\) 5518.55i 0.398163i 0.979983 + 0.199082i \(0.0637958\pi\)
−0.979983 + 0.199082i \(0.936204\pi\)
\(578\) 0 0
\(579\) 21295.2 + 12294.8i 1.52849 + 0.882475i
\(580\) 0 0
\(581\) 778.476 1348.36i 0.0555880 0.0962812i
\(582\) 0 0
\(583\) 19078.2 11014.8i 1.35530 0.782480i
\(584\) 0 0
\(585\) −10326.3 + 5418.68i −0.729813 + 0.382966i
\(586\) 0 0
\(587\) −13115.4 + 7572.16i −0.922196 + 0.532430i −0.884335 0.466853i \(-0.845388\pi\)
−0.0378611 + 0.999283i \(0.512054\pi\)
\(588\) 0 0
\(589\) −2478.51 + 4292.90i −0.173387 + 0.300315i
\(590\) 0 0
\(591\) −12531.8 7235.21i −0.872229 0.503582i
\(592\) 0 0
\(593\) 11663.1i 0.807667i 0.914832 + 0.403834i \(0.132323\pi\)
−0.914832 + 0.403834i \(0.867677\pi\)
\(594\) 0 0
\(595\) −560.146 970.201i −0.0385945 0.0668477i
\(596\) 0 0
\(597\) −4043.27 −0.277186
\(598\) 0 0
\(599\) 13655.9 0.931492 0.465746 0.884919i \(-0.345786\pi\)
0.465746 + 0.884919i \(0.345786\pi\)
\(600\) 0 0
\(601\) 11011.1 + 19071.8i 0.747341 + 1.29443i 0.949093 + 0.314996i \(0.102003\pi\)
−0.201752 + 0.979437i \(0.564663\pi\)
\(602\) 0 0
\(603\) 13139.0i 0.887331i
\(604\) 0 0
\(605\) 3185.78 + 1839.31i 0.214084 + 0.123601i
\(606\) 0 0
\(607\) −2114.49 + 3662.40i −0.141391 + 0.244896i −0.928021 0.372529i \(-0.878491\pi\)
0.786630 + 0.617425i \(0.211824\pi\)
\(608\) 0 0
\(609\) −1616.99 + 933.572i −0.107593 + 0.0621186i
\(610\) 0 0
\(611\) 6343.61 + 4012.12i 0.420024 + 0.265651i
\(612\) 0 0
\(613\) 23168.0 13376.1i 1.52651 0.881329i 0.527002 0.849864i \(-0.323316\pi\)
0.999505 0.0314651i \(-0.0100173\pi\)
\(614\) 0 0
\(615\) 449.953 779.341i 0.0295022 0.0510993i
\(616\) 0 0
\(617\) −1182.87 682.928i −0.0771806 0.0445602i 0.460913 0.887445i \(-0.347522\pi\)
−0.538094 + 0.842885i \(0.680855\pi\)
\(618\) 0 0
\(619\) 21506.1i 1.39645i 0.715879 + 0.698224i \(0.246026\pi\)
−0.715879 + 0.698224i \(0.753974\pi\)
\(620\) 0 0
\(621\) 3530.93 + 6115.74i 0.228166 + 0.395195i
\(622\) 0 0
\(623\) −1312.95 −0.0844340
\(624\) 0 0
\(625\) −18944.4 −1.21244
\(626\) 0 0
\(627\) 2460.34 + 4261.43i 0.156709 + 0.271428i
\(628\) 0 0
\(629\) 13280.8i 0.841878i
\(630\) 0 0
\(631\) −16347.6 9438.28i −1.03136 0.595455i −0.113985 0.993483i \(-0.536361\pi\)
−0.917373 + 0.398028i \(0.869695\pi\)
\(632\) 0 0
\(633\) −17056.5 + 29542.8i −1.07099 + 1.85501i
\(634\) 0 0
\(635\) −6567.75 + 3791.89i −0.410446 + 0.236971i
\(636\) 0 0
\(637\) −15730.3 635.092i −0.978425 0.0395028i
\(638\) 0 0
\(639\) 12150.5 7015.07i 0.752214 0.434291i
\(640\) 0 0
\(641\) −1819.89 + 3152.14i −0.112139 + 0.194231i −0.916633 0.399731i \(-0.869104\pi\)
0.804493 + 0.593962i \(0.202437\pi\)
\(642\) 0 0
\(643\) 21811.4 + 12592.8i 1.33773 + 0.772338i 0.986470 0.163941i \(-0.0524208\pi\)
0.351258 + 0.936279i \(0.385754\pi\)
\(644\) 0 0
\(645\) 17778.2i 1.08530i
\(646\) 0 0
\(647\) −4778.36 8276.36i −0.290350 0.502902i 0.683542 0.729911i \(-0.260439\pi\)
−0.973893 + 0.227009i \(0.927105\pi\)
\(648\) 0 0
\(649\) 16581.1 1.00287
\(650\) 0 0
\(651\) −5028.44 −0.302734
\(652\) 0 0
\(653\) −3020.59 5231.82i −0.181018 0.313533i 0.761209 0.648506i \(-0.224606\pi\)
−0.942228 + 0.334974i \(0.891273\pi\)
\(654\) 0 0
\(655\) 1713.49i 0.102216i
\(656\) 0 0
\(657\) −9488.57 5478.23i −0.563447 0.325306i
\(658\) 0 0
\(659\) 11127.9 19274.2i 0.657789 1.13932i −0.323398 0.946263i \(-0.604825\pi\)
0.981187 0.193061i \(-0.0618416\pi\)
\(660\) 0 0
\(661\) 24007.8 13860.9i 1.41270 0.815622i 0.417056 0.908881i \(-0.363062\pi\)
0.995642 + 0.0932592i \(0.0297285\pi\)
\(662\) 0 0
\(663\) 5609.20 8868.77i 0.328572 0.519509i
\(664\) 0 0
\(665\) 529.986 305.987i 0.0309052 0.0178431i
\(666\) 0 0
\(667\) −7058.24 + 12225.2i −0.409739 + 0.709689i
\(668\) 0 0
\(669\) −2785.37 1608.13i −0.160969 0.0929357i
\(670\) 0 0
\(671\) 24309.9i 1.39862i
\(672\) 0 0
\(673\) 1170.57 + 2027.48i 0.0670462 + 0.116127i 0.897600 0.440811i \(-0.145309\pi\)
−0.830554 + 0.556939i \(0.811976\pi\)
\(674\) 0 0
\(675\) 1964.45 0.112017
\(676\) 0 0
\(677\) −23282.1 −1.32172 −0.660859 0.750510i \(-0.729808\pi\)
−0.660859 + 0.750510i \(0.729808\pi\)
\(678\) 0 0
\(679\) −2278.57 3946.59i −0.128783 0.223058i
\(680\) 0 0
\(681\) 19724.5i 1.10990i
\(682\) 0 0
\(683\) −10508.4 6067.02i −0.588715 0.339895i 0.175874 0.984413i \(-0.443725\pi\)
−0.764589 + 0.644518i \(0.777058\pi\)
\(684\) 0 0
\(685\) −12871.1 + 22293.3i −0.717924 + 1.24348i
\(686\) 0 0
\(687\) −14428.8 + 8330.46i −0.801299 + 0.462630i
\(688\) 0 0
\(689\) 13717.8 21689.3i 0.758499 1.19927i
\(690\) 0 0
\(691\) −7712.28 + 4452.69i −0.424586 + 0.245135i −0.697038 0.717035i \(-0.745499\pi\)
0.272451 + 0.962170i \(0.412166\pi\)
\(692\) 0 0
\(693\) −1045.72 + 1811.24i −0.0573211 + 0.0992830i
\(694\) 0 0
\(695\) −29250.3 16887.7i −1.59644 0.921707i
\(696\) 0 0
\(697\) 339.290i 0.0184383i
\(698\) 0 0
\(699\) 9867.09 + 17090.3i 0.533916 + 0.924770i
\(700\) 0 0
\(701\) −6340.38 −0.341616 −0.170808 0.985304i \(-0.554638\pi\)
−0.170808 + 0.985304i \(0.554638\pi\)
\(702\) 0 0
\(703\) −7254.83 −0.389219
\(704\) 0 0
\(705\) 6974.41 + 12080.0i 0.372584 + 0.645334i
\(706\) 0 0
\(707\) 3246.16i 0.172679i
\(708\) 0 0
\(709\) −1858.25 1072.86i −0.0984316 0.0568295i 0.449976 0.893041i \(-0.351432\pi\)
−0.548408 + 0.836211i \(0.684766\pi\)
\(710\) 0 0
\(711\) 5721.73 9910.33i 0.301803 0.522738i
\(712\) 0 0
\(713\) −32924.0 + 19008.7i −1.72933 + 0.998430i
\(714\) 0 0
\(715\) 24078.5 + 972.141i 1.25942 + 0.0508476i
\(716\) 0 0
\(717\) 16644.0 9609.43i 0.866921 0.500517i
\(718\) 0 0
\(719\) −3993.78 + 6917.43i −0.207153 + 0.358799i −0.950817 0.309755i \(-0.899753\pi\)
0.743664 + 0.668554i \(0.233086\pi\)
\(720\) 0 0
\(721\) 1799.21 + 1038.77i 0.0929349 + 0.0536560i
\(722\) 0 0
\(723\) 307.339i 0.0158092i
\(724\) 0 0
\(725\) 1963.44 + 3400.79i 0.100580 + 0.174210i
\(726\) 0 0
\(727\) −10484.4 −0.534863 −0.267432 0.963577i \(-0.586175\pi\)
−0.267432 + 0.963577i \(0.586175\pi\)
\(728\) 0 0
\(729\) −7601.91 −0.386217
\(730\) 0 0
\(731\) −3351.45 5804.88i −0.169573 0.293709i
\(732\) 0 0
\(733\) 23183.2i 1.16820i −0.811682 0.584100i \(-0.801448\pi\)
0.811682 0.584100i \(-0.198552\pi\)
\(734\) 0 0
\(735\) −25337.0 14628.4i −1.27153 0.734115i
\(736\) 0 0
\(737\) 13575.4 23513.3i 0.678503 1.17520i
\(738\) 0 0
\(739\) 9036.24 5217.08i 0.449802 0.259693i −0.257945 0.966160i \(-0.583045\pi\)
0.707747 + 0.706466i \(0.249712\pi\)
\(740\) 0 0
\(741\) 4844.68 + 3064.10i 0.240181 + 0.151906i
\(742\) 0 0
\(743\) −6004.23 + 3466.54i −0.296465 + 0.171164i −0.640854 0.767663i \(-0.721420\pi\)
0.344389 + 0.938827i \(0.388086\pi\)
\(744\) 0 0
\(745\) 18030.7 31230.1i 0.886704 1.53582i
\(746\) 0 0
\(747\) 9834.33 + 5677.85i 0.481686 + 0.278101i
\(748\) 0 0
\(749\) 5246.58i 0.255949i
\(750\) 0 0
\(751\) 5623.54 + 9740.25i 0.273243 + 0.473271i 0.969690 0.244337i \(-0.0785703\pi\)
−0.696447 + 0.717608i \(0.745237\pi\)
\(752\) 0 0
\(753\) 40237.3 1.94732
\(754\) 0 0
\(755\) −40803.4 −1.96687
\(756\) 0 0
\(757\) −19825.8 34339.3i −0.951890 1.64872i −0.741331 0.671140i \(-0.765805\pi\)
−0.210559 0.977581i \(-0.567528\pi\)
\(758\) 0 0
\(759\) 37738.7i 1.80478i
\(760\) 0 0
\(761\) −1430.82 826.086i −0.0681567 0.0393503i 0.465534 0.885030i \(-0.345862\pi\)
−0.533691 + 0.845679i \(0.679195\pi\)
\(762\) 0 0
\(763\) 579.555 1003.82i 0.0274984 0.0476287i
\(764\) 0 0
\(765\) 7076.21 4085.45i 0.334433 0.193085i
\(766\) 0 0
\(767\) 17104.2 8975.34i 0.805212 0.422531i
\(768\) 0 0
\(769\) 8646.87 4992.27i 0.405480 0.234104i −0.283366 0.959012i \(-0.591451\pi\)
0.688846 + 0.724908i \(0.258118\pi\)
\(770\) 0 0
\(771\) 17525.3 30354.7i 0.818622 1.41790i
\(772\) 0 0
\(773\) −4550.65 2627.32i −0.211741 0.122248i 0.390379 0.920654i \(-0.372344\pi\)
−0.602120 + 0.798406i \(0.705677\pi\)
\(774\) 0 0
\(775\) 10575.6i 0.490176i
\(776\) 0 0
\(777\) −3679.68 6373.40i −0.169894 0.294266i
\(778\) 0 0
\(779\) −185.342 −0.00852446
\(780\) 0 0
\(781\) −28992.4 −1.32833
\(782\) 0 0
\(783\) 2632.93 + 4560.37i 0.120170 + 0.208141i
\(784\) 0 0
\(785\) 25910.0i 1.17805i
\(786\) 0 0
\(787\) 11671.4 + 6738.47i 0.528640 + 0.305210i 0.740462 0.672098i \(-0.234607\pi\)
−0.211823 + 0.977308i \(0.567940\pi\)
\(788\) 0 0
\(789\) 20951.2 36288.6i 0.945353 1.63740i
\(790\) 0 0
\(791\) 3192.24 1843.04i 0.143493 0.0828458i
\(792\) 0 0
\(793\) −13159.0 25076.9i −0.589266 1.12296i
\(794\) 0 0
\(795\) 41302.7 23846.1i 1.84258 1.06382i
\(796\) 0 0
\(797\) −16461.3 + 28511.7i −0.731603 + 1.26717i 0.224595 + 0.974452i \(0.427894\pi\)
−0.956198 + 0.292721i \(0.905439\pi\)
\(798\) 0 0
\(799\) −4554.51 2629.55i −0.201661 0.116429i
\(800\) 0 0
\(801\) 9576.09i 0.422415i
\(802\) 0 0
\(803\) 11320.4 + 19607.5i 0.497494 + 0.861686i
\(804\) 0 0
\(805\) 4693.49 0.205495
\(806\) 0 0
\(807\) −6392.68 −0.278851
\(808\) 0 0
\(809\) 2675.65 + 4634.36i 0.116280 + 0.201404i 0.918291 0.395906i \(-0.129570\pi\)
−0.802010 + 0.597310i \(0.796236\pi\)
\(810\) 0 0
\(811\) 26939.7i 1.16644i −0.812315 0.583219i \(-0.801793\pi\)
0.812315 0.583219i \(-0.198207\pi\)
\(812\) 0 0
\(813\) 10408.7 + 6009.48i 0.449016 + 0.259239i
\(814\) 0 0
\(815\) −3820.00 + 6616.44i −0.164183 + 0.284373i
\(816\) 0 0
\(817\) 3170.99 1830.77i 0.135788 0.0783974i
\(818\) 0 0
\(819\) −98.2872 + 2434.43i −0.00419345 + 0.103865i
\(820\) 0 0
\(821\) −30353.8 + 17524.8i −1.29032 + 0.744969i −0.978712 0.205240i \(-0.934203\pi\)
−0.311613 + 0.950209i \(0.600869\pi\)
\(822\) 0 0
\(823\) −12653.5 + 21916.6i −0.535935 + 0.928267i 0.463182 + 0.886263i \(0.346708\pi\)
−0.999117 + 0.0420043i \(0.986626\pi\)
\(824\) 0 0
\(825\) 9091.60 + 5249.04i 0.383671 + 0.221513i
\(826\) 0 0
\(827\) 1585.60i 0.0666709i 0.999444 + 0.0333355i \(0.0106130\pi\)
−0.999444 + 0.0333355i \(0.989387\pi\)
\(828\) 0 0
\(829\) 13274.2 + 22991.5i 0.556129 + 0.963244i 0.997815 + 0.0660740i \(0.0210473\pi\)
−0.441686 + 0.897170i \(0.645619\pi\)
\(830\) 0 0
\(831\) −44977.9 −1.87758
\(832\) 0 0
\(833\) 11030.6 0.458808
\(834\) 0 0
\(835\) 6632.41 + 11487.7i 0.274879 + 0.476104i
\(836\) 0 0
\(837\) 14181.6i 0.585648i
\(838\) 0 0
\(839\) 8588.06 + 4958.32i 0.353388 + 0.204029i 0.666177 0.745794i \(-0.267930\pi\)
−0.312788 + 0.949823i \(0.601263\pi\)
\(840\) 0 0
\(841\) 6931.34 12005.4i 0.284199 0.492248i
\(842\) 0 0
\(843\) −24578.1 + 14190.2i −1.00417 + 0.579757i
\(844\) 0 0
\(845\) 25364.4 12030.9i 1.03262 0.489793i
\(846\) 0 0
\(847\) 665.588 384.277i 0.0270010 0.0155891i
\(848\) 0 0
\(849\) −20166.8 + 34929.9i −0.815222 + 1.41201i
\(850\) 0 0
\(851\) −48185.9 27820.1i −1.94100 1.12064i
\(852\) 0 0
\(853\) 9649.84i 0.387344i 0.981066 + 0.193672i \(0.0620397\pi\)
−0.981066 + 0.193672i \(0.937960\pi\)
\(854\) 0 0
\(855\) 2231.73 + 3865.48i 0.0892674 + 0.154616i
\(856\) 0 0
\(857\) −24910.3 −0.992906 −0.496453 0.868063i \(-0.665365\pi\)
−0.496453 + 0.868063i \(0.665365\pi\)
\(858\) 0 0
\(859\) 24216.8 0.961892 0.480946 0.876750i \(-0.340293\pi\)
0.480946 + 0.876750i \(0.340293\pi\)
\(860\) 0 0
\(861\) −94.0061 162.823i −0.00372093 0.00644484i
\(862\) 0 0
\(863\) 35277.7i 1.39150i −0.718282 0.695752i \(-0.755071\pi\)
0.718282 0.695752i \(-0.244929\pi\)
\(864\) 0 0
\(865\) 12241.5 + 7067.63i 0.481183 + 0.277811i
\(866\) 0 0
\(867\) 13069.6 22637.2i 0.511957 0.886735i
\(868\) 0 0
\(869\) −20479.0 + 11823.6i −0.799429 + 0.461551i
\(870\) 0 0
\(871\) 1275.96 31603.5i 0.0496373 1.22944i
\(872\) 0 0
\(873\) 28784.7 16618.8i 1.11594 0.644287i
\(874\) 0 0
\(875\) −1479.19 + 2562.03i −0.0571495 + 0.0989858i
\(876\) 0 0
\(877\) −21877.6 12631.1i −0.842366 0.486340i 0.0157015 0.999877i \(-0.495002\pi\)
−0.858068 + 0.513536i \(0.828335\pi\)
\(878\) 0 0
\(879\) 46737.0i 1.79340i
\(880\) 0 0
\(881\) 10515.3 + 18213.1i 0.402124 + 0.696499i 0.993982 0.109543i \(-0.0349388\pi\)
−0.591858 + 0.806042i \(0.701605\pi\)
\(882\) 0 0
\(883\) 46166.1 1.75947 0.879736 0.475463i \(-0.157720\pi\)
0.879736 + 0.475463i \(0.157720\pi\)
\(884\) 0 0
\(885\) 35896.7 1.36345
\(886\) 0 0
\(887\) 1720.43 + 2979.87i 0.0651256 + 0.112801i 0.896750 0.442538i \(-0.145922\pi\)
−0.831624 + 0.555339i \(0.812589\pi\)
\(888\) 0 0
\(889\) 1584.44i 0.0597754i
\(890\) 0 0
\(891\) 30510.1 + 17615.0i 1.14717 + 0.662318i
\(892\) 0 0
\(893\) 1436.43 2487.96i 0.0538277 0.0932324i
\(894\) 0 0
\(895\) 17075.6 9858.59i 0.637736 0.368197i
\(896\) 0 0
\(897\) 20428.0 + 38929.4i 0.760392 + 1.44907i
\(898\) 0 0
\(899\) −24550.7 + 14174.3i −0.910801 + 0.525851i
\(900\) 0 0
\(901\) −8990.65 + 15572.3i −0.332433 + 0.575791i
\(902\) 0 0
\(903\) 3216.69 + 1857.15i 0.118543 + 0.0684410i
\(904\) 0 0
\(905\) 10392.9i 0.381735i
\(906\) 0 0
\(907\) 6298.16 + 10908.7i 0.230570 + 0.399359i 0.957976 0.286849i \(-0.0926077\pi\)
−0.727406 + 0.686207i \(0.759274\pi\)
\(908\) 0 0
\(909\) −23676.0 −0.863899
\(910\) 0 0
\(911\) 35459.3 1.28959 0.644797 0.764354i \(-0.276942\pi\)
0.644797 + 0.764354i \(0.276942\pi\)
\(912\) 0 0
\(913\) −11732.9 20322.0i −0.425304 0.736648i
\(914\) 0 0
\(915\) 52628.9i 1.90148i
\(916\) 0 0
\(917\) −310.028 178.995i −0.0111647 0.00644594i
\(918\) 0 0
\(919\) −13026.6 + 22562.8i −0.467584 + 0.809878i −0.999314 0.0370353i \(-0.988209\pi\)
0.531730 + 0.846914i \(0.321542\pi\)
\(920\) 0 0
\(921\) 16805.2 9702.49i 0.601249 0.347131i
\(922\) 0 0
\(923\) −29907.1 + 15693.6i −1.06653 + 0.559654i
\(924\) 0 0
\(925\) −13404.2 + 7738.94i −0.476463 + 0.275086i
\(926\) 0 0
\(927\) −7576.35 + 13122.6i −0.268436 + 0.464944i
\(928\) 0 0
\(929\) −13504.5 7796.81i −0.476929 0.275355i 0.242207 0.970225i \(-0.422129\pi\)
−0.719136 + 0.694869i \(0.755462\pi\)
\(930\) 0 0
\(931\) 6025.61i 0.212118i
\(932\) 0 0
\(933\) −20514.0 35531.4i −0.719828 1.24678i
\(934\) 0 0
\(935\) −16884.6 −0.590574
\(936\) 0 0
\(937\) −17686.2 −0.616631 −0.308316 0.951284i \(-0.599765\pi\)
−0.308316 + 0.951284i \(0.599765\pi\)
\(938\) 0 0
\(939\) 8988.79 + 15569.0i 0.312394 + 0.541082i
\(940\) 0 0
\(941\) 5601.91i 0.194067i 0.995281 + 0.0970336i \(0.0309354\pi\)
−0.995281 + 0.0970336i \(0.969065\pi\)
\(942\) 0 0
\(943\) −1231.02 710.731i −0.0425107 0.0245436i
\(944\) 0 0
\(945\) 875.404 1516.24i 0.0301343 0.0521941i
\(946\) 0 0
\(947\) −6347.18 + 3664.55i −0.217799 + 0.125746i −0.604931 0.796278i \(-0.706799\pi\)
0.387132 + 0.922024i \(0.373466\pi\)
\(948\) 0 0
\(949\) 22291.1 + 14098.4i 0.762487 + 0.482248i
\(950\) 0 0
\(951\) 53227.9 30731.2i 1.81497 1.04787i
\(952\) 0 0
\(953\) −19308.1 + 33442.6i −0.656296 + 1.13674i 0.325272 + 0.945621i \(0.394544\pi\)
−0.981567 + 0.191117i \(0.938789\pi\)
\(954\) 0 0
\(955\) 24781.5 + 14307.6i 0.839698 + 0.484800i
\(956\) 0 0
\(957\) 28140.9i 0.950539i
\(958\) 0 0
\(959\) 2689.08 + 4657.62i 0.0905473 + 0.156833i
\(960\) 0 0
\(961\) −46555.3 −1.56273
\(962\) 0 0
\(963\) −38266.2 −1.28049
\(964\) 0 0
\(965\) −23045.6 39916.2i −0.768773 1.33155i
\(966\) 0 0
\(967\) 26596.2i 0.884462i 0.896901 + 0.442231i \(0.145813\pi\)
−0.896901 + 0.442231i \(0.854187\pi\)
\(968\) 0 0
\(969\) −3478.33 2008.22i −0.115315 0.0665770i
\(970\) 0 0
\(971\) −15812.8 + 27388.6i −0.522613 + 0.905192i 0.477041 + 0.878881i \(0.341709\pi\)
−0.999654 + 0.0263111i \(0.991624\pi\)
\(972\) 0 0
\(973\) −6111.11 + 3528.25i −0.201349 + 0.116249i
\(974\) 0 0
\(975\) 12219.8 + 493.358i 0.401380 + 0.0162052i
\(976\) 0 0
\(977\) −28560.6 + 16489.5i −0.935246 + 0.539964i −0.888467 0.458941i \(-0.848229\pi\)
−0.0467788 + 0.998905i \(0.514896\pi\)
\(978\) 0 0
\(979\) −9894.18 + 17137.2i −0.323002 + 0.559457i
\(980\) 0 0
\(981\) 7321.40 + 4227.01i 0.238282 + 0.137572i
\(982\) 0 0
\(983\) 18103.5i 0.587397i 0.955898 + 0.293699i \(0.0948862\pi\)
−0.955898 + 0.293699i \(0.905114\pi\)
\(984\) 0 0
\(985\) 13561.9 + 23489.9i 0.438698 + 0.759847i
\(986\) 0 0
\(987\) 2914.25 0.0939834
\(988\) 0 0
\(989\) 28081.9 0.902885
\(990\) 0 0
\(991\) −13713.9 23753.2i −0.439593 0.761398i 0.558065 0.829797i \(-0.311544\pi\)
−0.997658 + 0.0683995i \(0.978211\pi\)
\(992\) 0 0
\(993\) 35084.1i 1.12121i
\(994\) 0 0
\(995\) 6563.44 + 3789.40i 0.209121 + 0.120736i
\(996\) 0 0
\(997\) −17249.5 + 29877.0i −0.547942 + 0.949063i 0.450474 + 0.892790i \(0.351255\pi\)
−0.998415 + 0.0562730i \(0.982078\pi\)
\(998\) 0 0
\(999\) −17974.7 + 10377.7i −0.569265 + 0.328665i
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 208.4.w.e.49.2 20
4.3 odd 2 104.4.o.a.49.9 yes 20
13.4 even 6 inner 208.4.w.e.17.2 20
52.11 even 12 1352.4.a.o.1.2 10
52.15 even 12 1352.4.a.p.1.2 10
52.43 odd 6 104.4.o.a.17.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.4.o.a.17.9 20 52.43 odd 6
104.4.o.a.49.9 yes 20 4.3 odd 2
208.4.w.e.17.2 20 13.4 even 6 inner
208.4.w.e.49.2 20 1.1 even 1 trivial
1352.4.a.o.1.2 10 52.11 even 12
1352.4.a.p.1.2 10 52.15 even 12