Properties

Label 252.4.x.a.209.24
Level $252$
Weight $4$
Character 252.209
Analytic conductor $14.868$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(41,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.x (of order \(6\), degree \(2\), 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: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 209.24
Character \(\chi\) \(=\) 252.209
Dual form 252.4.x.a.41.24

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.19208 - 0.205661i) q^{3} +(-2.34269 - 4.05766i) q^{5} +(18.4121 + 1.99859i) q^{7} +(26.9154 - 2.13562i) q^{9} +O(q^{10})\) \(q+(5.19208 - 0.205661i) q^{3} +(-2.34269 - 4.05766i) q^{5} +(18.4121 + 1.99859i) q^{7} +(26.9154 - 2.13562i) q^{9} +(16.1150 + 9.30397i) q^{11} +(-44.1430 + 25.4860i) q^{13} +(-12.9980 - 20.5859i) q^{15} +112.833 q^{17} -111.459i q^{19} +(96.0082 + 6.59018i) q^{21} +(-124.503 + 71.8819i) q^{23} +(51.5236 - 89.2414i) q^{25} +(139.308 - 16.6238i) q^{27} +(206.907 + 119.458i) q^{29} +(179.999 - 103.923i) q^{31} +(85.5836 + 44.9928i) q^{33} +(-35.0243 - 79.3922i) q^{35} -227.815 q^{37} +(-223.953 + 141.404i) q^{39} +(-133.141 - 230.607i) q^{41} +(-170.287 + 294.946i) q^{43} +(-71.7202 - 104.211i) q^{45} +(111.979 - 193.952i) q^{47} +(335.011 + 73.5964i) q^{49} +(585.836 - 23.2053i) q^{51} +547.974i q^{53} -87.1854i q^{55} +(-22.9227 - 578.703i) q^{57} +(43.9483 + 76.1207i) q^{59} +(312.398 + 180.363i) q^{61} +(499.838 + 14.4716i) q^{63} +(206.827 + 119.412i) q^{65} +(-372.426 - 645.060i) q^{67} +(-631.647 + 398.822i) q^{69} +135.948i q^{71} -467.289i q^{73} +(249.161 - 473.945i) q^{75} +(278.115 + 203.513i) q^{77} +(192.171 - 332.850i) q^{79} +(719.878 - 114.962i) q^{81} +(-597.216 + 1034.41i) q^{83} +(-264.332 - 457.836i) q^{85} +(1098.85 + 577.683i) q^{87} -1385.63 q^{89} +(-863.702 + 381.027i) q^{91} +(913.198 - 576.594i) q^{93} +(-452.262 + 261.114i) q^{95} +(-1070.10 - 617.821i) q^{97} +(453.610 + 216.005i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 6 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 6 q^{7} + 60 q^{9} - 12 q^{11} + 192 q^{15} - 72 q^{21} - 408 q^{23} - 600 q^{25} - 84 q^{29} + 336 q^{37} + 36 q^{39} + 84 q^{43} + 318 q^{49} - 1812 q^{51} - 852 q^{57} - 564 q^{63} + 2964 q^{65} - 588 q^{67} + 2400 q^{77} + 204 q^{79} + 1980 q^{81} - 360 q^{85} - 1080 q^{91} + 2496 q^{93} + 300 q^{95} - 4968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

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
\(3\) 5.19208 0.205661i 0.999216 0.0395795i
\(4\) 0 0
\(5\) −2.34269 4.05766i −0.209537 0.362929i 0.742032 0.670365i \(-0.233862\pi\)
−0.951569 + 0.307436i \(0.900529\pi\)
\(6\) 0 0
\(7\) 18.4121 + 1.99859i 0.994160 + 0.107914i
\(8\) 0 0
\(9\) 26.9154 2.13562i 0.996867 0.0790970i
\(10\) 0 0
\(11\) 16.1150 + 9.30397i 0.441713 + 0.255023i 0.704324 0.709879i \(-0.251250\pi\)
−0.262611 + 0.964902i \(0.584584\pi\)
\(12\) 0 0
\(13\) −44.1430 + 25.4860i −0.941775 + 0.543734i −0.890516 0.454951i \(-0.849657\pi\)
−0.0512589 + 0.998685i \(0.516323\pi\)
\(14\) 0 0
\(15\) −12.9980 20.5859i −0.223737 0.354351i
\(16\) 0 0
\(17\) 112.833 1.60976 0.804880 0.593438i \(-0.202230\pi\)
0.804880 + 0.593438i \(0.202230\pi\)
\(18\) 0 0
\(19\) 111.459i 1.34581i −0.739729 0.672905i \(-0.765046\pi\)
0.739729 0.672905i \(-0.234954\pi\)
\(20\) 0 0
\(21\) 96.0082 + 6.59018i 0.997652 + 0.0684807i
\(22\) 0 0
\(23\) −124.503 + 71.8819i −1.12873 + 0.651671i −0.943614 0.331047i \(-0.892598\pi\)
−0.185112 + 0.982717i \(0.559265\pi\)
\(24\) 0 0
\(25\) 51.5236 89.2414i 0.412189 0.713932i
\(26\) 0 0
\(27\) 139.308 16.6238i 0.992955 0.118490i
\(28\) 0 0
\(29\) 206.907 + 119.458i 1.32489 + 0.764924i 0.984504 0.175363i \(-0.0561101\pi\)
0.340383 + 0.940287i \(0.389443\pi\)
\(30\) 0 0
\(31\) 179.999 103.923i 1.04287 0.602099i 0.122222 0.992503i \(-0.460998\pi\)
0.920644 + 0.390404i \(0.127665\pi\)
\(32\) 0 0
\(33\) 85.5836 + 44.9928i 0.451460 + 0.237340i
\(34\) 0 0
\(35\) −35.0243 79.3922i −0.169148 0.383421i
\(36\) 0 0
\(37\) −227.815 −1.01223 −0.506115 0.862466i \(-0.668919\pi\)
−0.506115 + 0.862466i \(0.668919\pi\)
\(38\) 0 0
\(39\) −223.953 + 141.404i −0.919517 + 0.580583i
\(40\) 0 0
\(41\) −133.141 230.607i −0.507149 0.878407i −0.999966 0.00827417i \(-0.997366\pi\)
0.492817 0.870133i \(-0.335967\pi\)
\(42\) 0 0
\(43\) −170.287 + 294.946i −0.603920 + 1.04602i 0.388301 + 0.921533i \(0.373062\pi\)
−0.992221 + 0.124488i \(0.960271\pi\)
\(44\) 0 0
\(45\) −71.7202 104.211i −0.237587 0.345218i
\(46\) 0 0
\(47\) 111.979 193.952i 0.347527 0.601934i −0.638283 0.769802i \(-0.720355\pi\)
0.985809 + 0.167868i \(0.0536883\pi\)
\(48\) 0 0
\(49\) 335.011 + 73.5964i 0.976709 + 0.214567i
\(50\) 0 0
\(51\) 585.836 23.2053i 1.60850 0.0637135i
\(52\) 0 0
\(53\) 547.974i 1.42019i 0.704107 + 0.710094i \(0.251348\pi\)
−0.704107 + 0.710094i \(0.748652\pi\)
\(54\) 0 0
\(55\) 87.1854i 0.213747i
\(56\) 0 0
\(57\) −22.9227 578.703i −0.0532665 1.34476i
\(58\) 0 0
\(59\) 43.9483 + 76.1207i 0.0969760 + 0.167967i 0.910432 0.413660i \(-0.135750\pi\)
−0.813456 + 0.581627i \(0.802416\pi\)
\(60\) 0 0
\(61\) 312.398 + 180.363i 0.655712 + 0.378575i 0.790641 0.612280i \(-0.209747\pi\)
−0.134929 + 0.990855i \(0.543081\pi\)
\(62\) 0 0
\(63\) 499.838 + 14.4716i 0.999581 + 0.0289404i
\(64\) 0 0
\(65\) 206.827 + 119.412i 0.394673 + 0.227865i
\(66\) 0 0
\(67\) −372.426 645.060i −0.679090 1.17622i −0.975255 0.221082i \(-0.929041\pi\)
0.296165 0.955137i \(-0.404292\pi\)
\(68\) 0 0
\(69\) −631.647 + 398.822i −1.10205 + 0.695834i
\(70\) 0 0
\(71\) 135.948i 0.227240i 0.993524 + 0.113620i \(0.0362447\pi\)
−0.993524 + 0.113620i \(0.963755\pi\)
\(72\) 0 0
\(73\) 467.289i 0.749206i −0.927185 0.374603i \(-0.877779\pi\)
0.927185 0.374603i \(-0.122221\pi\)
\(74\) 0 0
\(75\) 249.161 473.945i 0.383609 0.729686i
\(76\) 0 0
\(77\) 278.115 + 203.513i 0.411613 + 0.301201i
\(78\) 0 0
\(79\) 192.171 332.850i 0.273683 0.474032i −0.696119 0.717926i \(-0.745092\pi\)
0.969802 + 0.243894i \(0.0784249\pi\)
\(80\) 0 0
\(81\) 719.878 114.962i 0.987487 0.157698i
\(82\) 0 0
\(83\) −597.216 + 1034.41i −0.789795 + 1.36796i 0.136297 + 0.990668i \(0.456480\pi\)
−0.926092 + 0.377297i \(0.876854\pi\)
\(84\) 0 0
\(85\) −264.332 457.836i −0.337304 0.584228i
\(86\) 0 0
\(87\) 1098.85 + 577.683i 1.35412 + 0.711886i
\(88\) 0 0
\(89\) −1385.63 −1.65030 −0.825150 0.564914i \(-0.808909\pi\)
−0.825150 + 0.564914i \(0.808909\pi\)
\(90\) 0 0
\(91\) −863.702 + 381.027i −0.994952 + 0.438929i
\(92\) 0 0
\(93\) 913.198 576.594i 1.01822 0.642903i
\(94\) 0 0
\(95\) −452.262 + 261.114i −0.488433 + 0.281997i
\(96\) 0 0
\(97\) −1070.10 617.821i −1.12012 0.646704i −0.178691 0.983905i \(-0.557186\pi\)
−0.941432 + 0.337202i \(0.890520\pi\)
\(98\) 0 0
\(99\) 453.610 + 216.005i 0.460501 + 0.219286i
\(100\) 0 0
\(101\) 94.1707 163.108i 0.0927756 0.160692i −0.815902 0.578190i \(-0.803759\pi\)
0.908678 + 0.417498i \(0.137093\pi\)
\(102\) 0 0
\(103\) −687.152 + 396.727i −0.657350 + 0.379521i −0.791267 0.611471i \(-0.790578\pi\)
0.133916 + 0.990993i \(0.457245\pi\)
\(104\) 0 0
\(105\) −198.177 405.008i −0.184191 0.376426i
\(106\) 0 0
\(107\) 1399.55i 1.26448i 0.774773 + 0.632240i \(0.217864\pi\)
−0.774773 + 0.632240i \(0.782136\pi\)
\(108\) 0 0
\(109\) −2137.48 −1.87829 −0.939145 0.343521i \(-0.888380\pi\)
−0.939145 + 0.343521i \(0.888380\pi\)
\(110\) 0 0
\(111\) −1182.83 + 46.8526i −1.01144 + 0.0400635i
\(112\) 0 0
\(113\) −1500.15 + 866.110i −1.24887 + 0.721034i −0.970884 0.239552i \(-0.922999\pi\)
−0.277984 + 0.960586i \(0.589666\pi\)
\(114\) 0 0
\(115\) 583.346 + 336.795i 0.473020 + 0.273098i
\(116\) 0 0
\(117\) −1133.70 + 780.239i −0.895817 + 0.616522i
\(118\) 0 0
\(119\) 2077.48 + 225.506i 1.60036 + 0.173715i
\(120\) 0 0
\(121\) −492.372 852.814i −0.369926 0.640731i
\(122\) 0 0
\(123\) −738.704 1169.95i −0.541518 0.857646i
\(124\) 0 0
\(125\) −1068.49 −0.764549
\(126\) 0 0
\(127\) −770.489 −0.538345 −0.269173 0.963092i \(-0.586750\pi\)
−0.269173 + 0.963092i \(0.586750\pi\)
\(128\) 0 0
\(129\) −823.487 + 1566.41i −0.562046 + 1.06910i
\(130\) 0 0
\(131\) 489.800 + 848.359i 0.326672 + 0.565813i 0.981849 0.189662i \(-0.0607393\pi\)
−0.655177 + 0.755475i \(0.727406\pi\)
\(132\) 0 0
\(133\) 222.760 2052.19i 0.145231 1.33795i
\(134\) 0 0
\(135\) −393.809 526.320i −0.251064 0.335544i
\(136\) 0 0
\(137\) −182.458 105.342i −0.113784 0.0656932i 0.442028 0.897001i \(-0.354259\pi\)
−0.555812 + 0.831308i \(0.687593\pi\)
\(138\) 0 0
\(139\) 730.515 421.763i 0.445766 0.257363i −0.260274 0.965535i \(-0.583813\pi\)
0.706040 + 0.708172i \(0.250480\pi\)
\(140\) 0 0
\(141\) 541.513 1030.05i 0.323430 0.615217i
\(142\) 0 0
\(143\) −948.484 −0.554659
\(144\) 0 0
\(145\) 1119.41i 0.641119i
\(146\) 0 0
\(147\) 1754.54 + 313.220i 0.984436 + 0.175741i
\(148\) 0 0
\(149\) 1137.32 656.632i 0.625321 0.361029i −0.153617 0.988131i \(-0.549092\pi\)
0.778938 + 0.627101i \(0.215759\pi\)
\(150\) 0 0
\(151\) −1738.76 + 3011.62i −0.937074 + 1.62306i −0.166182 + 0.986095i \(0.553144\pi\)
−0.770893 + 0.636965i \(0.780190\pi\)
\(152\) 0 0
\(153\) 3036.93 240.967i 1.60472 0.127327i
\(154\) 0 0
\(155\) −843.366 486.918i −0.437038 0.252324i
\(156\) 0 0
\(157\) −499.729 + 288.519i −0.254030 + 0.146664i −0.621608 0.783328i \(-0.713520\pi\)
0.367578 + 0.929993i \(0.380187\pi\)
\(158\) 0 0
\(159\) 112.697 + 2845.12i 0.0562103 + 1.41908i
\(160\) 0 0
\(161\) −2436.03 + 1074.67i −1.19246 + 0.526060i
\(162\) 0 0
\(163\) 1595.73 0.766794 0.383397 0.923584i \(-0.374754\pi\)
0.383397 + 0.923584i \(0.374754\pi\)
\(164\) 0 0
\(165\) −17.9307 452.674i −0.00846000 0.213579i
\(166\) 0 0
\(167\) −184.918 320.287i −0.0856850 0.148411i 0.819998 0.572366i \(-0.193975\pi\)
−0.905683 + 0.423956i \(0.860641\pi\)
\(168\) 0 0
\(169\) 200.572 347.402i 0.0912938 0.158125i
\(170\) 0 0
\(171\) −238.033 2999.96i −0.106450 1.34159i
\(172\) 0 0
\(173\) 973.579 1686.29i 0.427860 0.741076i −0.568822 0.822460i \(-0.692601\pi\)
0.996683 + 0.0813844i \(0.0259341\pi\)
\(174\) 0 0
\(175\) 1127.01 1540.15i 0.486824 0.665282i
\(176\) 0 0
\(177\) 243.838 + 386.187i 0.103548 + 0.163997i
\(178\) 0 0
\(179\) 851.921i 0.355730i −0.984055 0.177865i \(-0.943081\pi\)
0.984055 0.177865i \(-0.0569190\pi\)
\(180\) 0 0
\(181\) 639.259i 0.262518i 0.991348 + 0.131259i \(0.0419020\pi\)
−0.991348 + 0.131259i \(0.958098\pi\)
\(182\) 0 0
\(183\) 1659.09 + 872.210i 0.670182 + 0.352326i
\(184\) 0 0
\(185\) 533.700 + 924.395i 0.212099 + 0.367367i
\(186\) 0 0
\(187\) 1818.29 + 1049.79i 0.711051 + 0.410526i
\(188\) 0 0
\(189\) 2598.17 27.6595i 0.999943 0.0106451i
\(190\) 0 0
\(191\) 3120.13 + 1801.41i 1.18201 + 0.682435i 0.956479 0.291800i \(-0.0942542\pi\)
0.225533 + 0.974235i \(0.427588\pi\)
\(192\) 0 0
\(193\) −1745.01 3022.44i −0.650821 1.12725i −0.982924 0.184012i \(-0.941092\pi\)
0.332103 0.943243i \(-0.392242\pi\)
\(194\) 0 0
\(195\) 1098.42 + 577.459i 0.403383 + 0.212065i
\(196\) 0 0
\(197\) 1909.86i 0.690719i 0.938470 + 0.345360i \(0.112243\pi\)
−0.938470 + 0.345360i \(0.887757\pi\)
\(198\) 0 0
\(199\) 713.005i 0.253988i −0.991903 0.126994i \(-0.959467\pi\)
0.991903 0.126994i \(-0.0405329\pi\)
\(200\) 0 0
\(201\) −2066.33 3272.61i −0.725112 1.14842i
\(202\) 0 0
\(203\) 3570.85 + 2612.99i 1.23460 + 0.903430i
\(204\) 0 0
\(205\) −623.816 + 1080.48i −0.212533 + 0.368117i
\(206\) 0 0
\(207\) −3197.54 + 2200.62i −1.07364 + 0.738908i
\(208\) 0 0
\(209\) 1037.01 1796.15i 0.343213 0.594462i
\(210\) 0 0
\(211\) 788.596 + 1365.89i 0.257295 + 0.445648i 0.965516 0.260343i \(-0.0838355\pi\)
−0.708221 + 0.705990i \(0.750502\pi\)
\(212\) 0 0
\(213\) 27.9592 + 705.853i 0.00899406 + 0.227062i
\(214\) 0 0
\(215\) 1595.72 0.506174
\(216\) 0 0
\(217\) 3521.86 1553.69i 1.10175 0.486043i
\(218\) 0 0
\(219\) −96.1031 2426.20i −0.0296532 0.748618i
\(220\) 0 0
\(221\) −4980.77 + 2875.65i −1.51603 + 0.875281i
\(222\) 0 0
\(223\) −412.569 238.197i −0.123891 0.0715284i 0.436774 0.899571i \(-0.356121\pi\)
−0.560665 + 0.828043i \(0.689454\pi\)
\(224\) 0 0
\(225\) 1196.19 2512.00i 0.354427 0.744298i
\(226\) 0 0
\(227\) 1386.85 2402.10i 0.405501 0.702348i −0.588879 0.808221i \(-0.700431\pi\)
0.994380 + 0.105873i \(0.0337639\pi\)
\(228\) 0 0
\(229\) 1698.53 980.646i 0.490139 0.282982i −0.234493 0.972118i \(-0.575343\pi\)
0.724632 + 0.689136i \(0.242010\pi\)
\(230\) 0 0
\(231\) 1485.85 + 999.458i 0.423212 + 0.284673i
\(232\) 0 0
\(233\) 4734.94i 1.33132i −0.746257 0.665658i \(-0.768151\pi\)
0.746257 0.665658i \(-0.231849\pi\)
\(234\) 0 0
\(235\) −1049.33 −0.291279
\(236\) 0 0
\(237\) 929.313 1767.71i 0.254706 0.484493i
\(238\) 0 0
\(239\) −1055.48 + 609.379i −0.285661 + 0.164927i −0.635984 0.771703i \(-0.719405\pi\)
0.350322 + 0.936629i \(0.386072\pi\)
\(240\) 0 0
\(241\) 1784.77 + 1030.43i 0.477041 + 0.275420i 0.719182 0.694821i \(-0.244517\pi\)
−0.242142 + 0.970241i \(0.577850\pi\)
\(242\) 0 0
\(243\) 3714.02 744.943i 0.980472 0.196659i
\(244\) 0 0
\(245\) −486.199 1531.78i −0.126784 0.399435i
\(246\) 0 0
\(247\) 2840.64 + 4920.13i 0.731763 + 1.26745i
\(248\) 0 0
\(249\) −2888.06 + 5493.56i −0.735033 + 1.39815i
\(250\) 0 0
\(251\) 2966.29 0.745937 0.372969 0.927844i \(-0.378340\pi\)
0.372969 + 0.927844i \(0.378340\pi\)
\(252\) 0 0
\(253\) −2675.15 −0.664764
\(254\) 0 0
\(255\) −1466.59 2322.76i −0.360163 0.570419i
\(256\) 0 0
\(257\) −1133.09 1962.58i −0.275021 0.476351i 0.695119 0.718895i \(-0.255352\pi\)
−0.970141 + 0.242543i \(0.922018\pi\)
\(258\) 0 0
\(259\) −4194.54 455.307i −1.00632 0.109233i
\(260\) 0 0
\(261\) 5824.11 + 2773.38i 1.38124 + 0.657732i
\(262\) 0 0
\(263\) −1580.16 912.308i −0.370483 0.213898i 0.303186 0.952931i \(-0.401950\pi\)
−0.673669 + 0.739033i \(0.735283\pi\)
\(264\) 0 0
\(265\) 2223.49 1283.73i 0.515427 0.297582i
\(266\) 0 0
\(267\) −7194.31 + 284.971i −1.64901 + 0.0653180i
\(268\) 0 0
\(269\) −5616.89 −1.27312 −0.636558 0.771229i \(-0.719642\pi\)
−0.636558 + 0.771229i \(0.719642\pi\)
\(270\) 0 0
\(271\) 5603.22i 1.25598i 0.778220 + 0.627992i \(0.216123\pi\)
−0.778220 + 0.627992i \(0.783877\pi\)
\(272\) 0 0
\(273\) −4406.05 + 2155.95i −0.976800 + 0.477964i
\(274\) 0 0
\(275\) 1660.60 958.748i 0.364138 0.210235i
\(276\) 0 0
\(277\) 2329.35 4034.55i 0.505260 0.875137i −0.494721 0.869052i \(-0.664730\pi\)
0.999981 0.00608483i \(-0.00193687\pi\)
\(278\) 0 0
\(279\) 4622.82 3181.53i 0.991974 0.682700i
\(280\) 0 0
\(281\) 1939.85 + 1119.97i 0.411820 + 0.237765i 0.691572 0.722308i \(-0.256919\pi\)
−0.279751 + 0.960073i \(0.590252\pi\)
\(282\) 0 0
\(283\) −2759.51 + 1593.21i −0.579633 + 0.334651i −0.760987 0.648767i \(-0.775285\pi\)
0.181355 + 0.983418i \(0.441952\pi\)
\(284\) 0 0
\(285\) −2294.48 + 1448.74i −0.476889 + 0.301108i
\(286\) 0 0
\(287\) −1990.51 4512.05i −0.409395 0.928006i
\(288\) 0 0
\(289\) 7818.18 1.59132
\(290\) 0 0
\(291\) −5683.10 2987.70i −1.14484 0.601863i
\(292\) 0 0
\(293\) 4563.57 + 7904.34i 0.909921 + 1.57603i 0.814172 + 0.580624i \(0.197191\pi\)
0.0957489 + 0.995406i \(0.469475\pi\)
\(294\) 0 0
\(295\) 205.915 356.655i 0.0406401 0.0703907i
\(296\) 0 0
\(297\) 2399.61 + 1028.22i 0.468819 + 0.200888i
\(298\) 0 0
\(299\) 3663.97 6346.18i 0.708671 1.22745i
\(300\) 0 0
\(301\) −3724.82 + 5090.25i −0.713274 + 0.974741i
\(302\) 0 0
\(303\) 455.397 866.239i 0.0863428 0.164238i
\(304\) 0 0
\(305\) 1690.14i 0.317302i
\(306\) 0 0
\(307\) 4744.58i 0.882044i −0.897496 0.441022i \(-0.854616\pi\)
0.897496 0.441022i \(-0.145384\pi\)
\(308\) 0 0
\(309\) −3486.16 + 2201.16i −0.641814 + 0.405242i
\(310\) 0 0
\(311\) 1089.62 + 1887.29i 0.198672 + 0.344110i 0.948098 0.317978i \(-0.103004\pi\)
−0.749426 + 0.662088i \(0.769671\pi\)
\(312\) 0 0
\(313\) 1752.85 + 1012.01i 0.316540 + 0.182754i 0.649849 0.760063i \(-0.274832\pi\)
−0.333309 + 0.942818i \(0.608165\pi\)
\(314\) 0 0
\(315\) −1112.25 2062.08i −0.198946 0.368841i
\(316\) 0 0
\(317\) 189.948 + 109.666i 0.0336547 + 0.0194305i 0.516733 0.856147i \(-0.327148\pi\)
−0.483078 + 0.875577i \(0.660481\pi\)
\(318\) 0 0
\(319\) 2222.87 + 3850.12i 0.390146 + 0.675753i
\(320\) 0 0
\(321\) 287.833 + 7266.56i 0.0500475 + 1.26349i
\(322\) 0 0
\(323\) 12576.2i 2.16643i
\(324\) 0 0
\(325\) 5252.52i 0.896484i
\(326\) 0 0
\(327\) −11098.0 + 439.597i −1.87682 + 0.0743418i
\(328\) 0 0
\(329\) 2449.39 3347.27i 0.410454 0.560916i
\(330\) 0 0
\(331\) 1884.36 3263.81i 0.312912 0.541980i −0.666079 0.745881i \(-0.732029\pi\)
0.978991 + 0.203901i \(0.0653621\pi\)
\(332\) 0 0
\(333\) −6131.72 + 486.525i −1.00906 + 0.0800643i
\(334\) 0 0
\(335\) −1744.96 + 3022.36i −0.284589 + 0.492922i
\(336\) 0 0
\(337\) 3056.60 + 5294.18i 0.494076 + 0.855764i 0.999977 0.00682736i \(-0.00217323\pi\)
−0.505901 + 0.862592i \(0.668840\pi\)
\(338\) 0 0
\(339\) −7610.76 + 4805.44i −1.21935 + 0.769898i
\(340\) 0 0
\(341\) 3867.57 0.614196
\(342\) 0 0
\(343\) 6021.17 + 2024.61i 0.947851 + 0.318714i
\(344\) 0 0
\(345\) 3098.04 + 1628.69i 0.483458 + 0.254162i
\(346\) 0 0
\(347\) 1143.36 660.117i 0.176883 0.102124i −0.408944 0.912559i \(-0.634103\pi\)
0.585828 + 0.810436i \(0.300770\pi\)
\(348\) 0 0
\(349\) 2309.07 + 1333.14i 0.354159 + 0.204474i 0.666515 0.745491i \(-0.267785\pi\)
−0.312357 + 0.949965i \(0.601118\pi\)
\(350\) 0 0
\(351\) −5725.80 + 4284.22i −0.870713 + 0.651495i
\(352\) 0 0
\(353\) 4725.05 8184.03i 0.712434 1.23397i −0.251508 0.967855i \(-0.580926\pi\)
0.963941 0.266116i \(-0.0857403\pi\)
\(354\) 0 0
\(355\) 551.632 318.485i 0.0824720 0.0476152i
\(356\) 0 0
\(357\) 10832.8 + 743.586i 1.60598 + 0.110237i
\(358\) 0 0
\(359\) 6846.88i 1.00659i −0.864116 0.503293i \(-0.832122\pi\)
0.864116 0.503293i \(-0.167878\pi\)
\(360\) 0 0
\(361\) −5564.07 −0.811206
\(362\) 0 0
\(363\) −2731.83 4326.62i −0.394996 0.625588i
\(364\) 0 0
\(365\) −1896.10 + 1094.71i −0.271908 + 0.156986i
\(366\) 0 0
\(367\) −5440.63 3141.15i −0.773838 0.446775i 0.0604042 0.998174i \(-0.480761\pi\)
−0.834242 + 0.551399i \(0.814094\pi\)
\(368\) 0 0
\(369\) −4076.02 5922.53i −0.575039 0.835541i
\(370\) 0 0
\(371\) −1095.17 + 10089.4i −0.153258 + 1.41189i
\(372\) 0 0
\(373\) −5253.75 9099.76i −0.729300 1.26318i −0.957179 0.289495i \(-0.906513\pi\)
0.227880 0.973689i \(-0.426821\pi\)
\(374\) 0 0
\(375\) −5547.68 + 219.747i −0.763950 + 0.0302605i
\(376\) 0 0
\(377\) −12178.0 −1.66366
\(378\) 0 0
\(379\) 5904.89 0.800301 0.400150 0.916450i \(-0.368958\pi\)
0.400150 + 0.916450i \(0.368958\pi\)
\(380\) 0 0
\(381\) −4000.44 + 158.460i −0.537923 + 0.0213074i
\(382\) 0 0
\(383\) 1175.26 + 2035.61i 0.156797 + 0.271580i 0.933712 0.358026i \(-0.116550\pi\)
−0.776915 + 0.629605i \(0.783217\pi\)
\(384\) 0 0
\(385\) 174.248 1605.27i 0.0230662 0.212499i
\(386\) 0 0
\(387\) −3953.46 + 8302.27i −0.519291 + 1.09051i
\(388\) 0 0
\(389\) −11034.6 6370.81i −1.43824 0.830368i −0.440511 0.897747i \(-0.645203\pi\)
−0.997727 + 0.0673794i \(0.978536\pi\)
\(390\) 0 0
\(391\) −14048.0 + 8110.62i −1.81698 + 1.04903i
\(392\) 0 0
\(393\) 2717.56 + 4304.02i 0.348811 + 0.552440i
\(394\) 0 0
\(395\) −1800.79 −0.229386
\(396\) 0 0
\(397\) 13326.3i 1.68470i 0.538930 + 0.842350i \(0.318829\pi\)
−0.538930 + 0.842350i \(0.681171\pi\)
\(398\) 0 0
\(399\) 734.533 10701.0i 0.0921621 1.34265i
\(400\) 0 0
\(401\) −8090.75 + 4671.20i −1.00756 + 0.581717i −0.910477 0.413559i \(-0.864285\pi\)
−0.0970856 + 0.995276i \(0.530952\pi\)
\(402\) 0 0
\(403\) −5297.15 + 9174.92i −0.654763 + 1.13408i
\(404\) 0 0
\(405\) −2152.93 2651.70i −0.264148 0.325344i
\(406\) 0 0
\(407\) −3671.22 2119.58i −0.447115 0.258142i
\(408\) 0 0
\(409\) −268.033 + 154.749i −0.0324044 + 0.0187087i −0.516115 0.856519i \(-0.672622\pi\)
0.483710 + 0.875228i \(0.339289\pi\)
\(410\) 0 0
\(411\) −968.999 509.419i −0.116295 0.0611382i
\(412\) 0 0
\(413\) 657.047 + 1489.38i 0.0782837 + 0.177452i
\(414\) 0 0
\(415\) 5596.38 0.661965
\(416\) 0 0
\(417\) 3706.15 2340.07i 0.435230 0.274805i
\(418\) 0 0
\(419\) 3008.72 + 5211.26i 0.350801 + 0.607605i 0.986390 0.164422i \(-0.0525760\pi\)
−0.635589 + 0.772028i \(0.719243\pi\)
\(420\) 0 0
\(421\) 1920.18 3325.85i 0.222289 0.385017i −0.733213 0.679999i \(-0.761980\pi\)
0.955503 + 0.294982i \(0.0953137\pi\)
\(422\) 0 0
\(423\) 2599.74 5459.45i 0.298827 0.627536i
\(424\) 0 0
\(425\) 5813.53 10069.3i 0.663524 1.14926i
\(426\) 0 0
\(427\) 5391.43 + 3945.21i 0.611029 + 0.447125i
\(428\) 0 0
\(429\) −4924.61 + 195.066i −0.554224 + 0.0219531i
\(430\) 0 0
\(431\) 4510.80i 0.504125i −0.967711 0.252062i \(-0.918891\pi\)
0.967711 0.252062i \(-0.0811088\pi\)
\(432\) 0 0
\(433\) 7012.84i 0.778328i −0.921169 0.389164i \(-0.872764\pi\)
0.921169 0.389164i \(-0.127236\pi\)
\(434\) 0 0
\(435\) −230.220 5812.09i −0.0253752 0.640616i
\(436\) 0 0
\(437\) 8011.88 + 13877.0i 0.877025 + 1.51905i
\(438\) 0 0
\(439\) 3220.45 + 1859.33i 0.350122 + 0.202143i 0.664739 0.747075i \(-0.268543\pi\)
−0.314617 + 0.949219i \(0.601876\pi\)
\(440\) 0 0
\(441\) 9174.14 + 1265.42i 0.990621 + 0.136640i
\(442\) 0 0
\(443\) 5177.61 + 2989.29i 0.555295 + 0.320599i 0.751255 0.660012i \(-0.229449\pi\)
−0.195960 + 0.980612i \(0.562782\pi\)
\(444\) 0 0
\(445\) 3246.11 + 5622.43i 0.345799 + 0.598941i
\(446\) 0 0
\(447\) 5770.01 3643.19i 0.610542 0.385496i
\(448\) 0 0
\(449\) 1188.91i 0.124963i −0.998046 0.0624813i \(-0.980099\pi\)
0.998046 0.0624813i \(-0.0199014\pi\)
\(450\) 0 0
\(451\) 4954.95i 0.517338i
\(452\) 0 0
\(453\) −8408.41 + 15994.2i −0.872100 + 1.65888i
\(454\) 0 0
\(455\) 3569.47 + 2611.98i 0.367779 + 0.269125i
\(456\) 0 0
\(457\) 2039.30 3532.17i 0.208741 0.361549i −0.742577 0.669760i \(-0.766397\pi\)
0.951318 + 0.308211i \(0.0997302\pi\)
\(458\) 0 0
\(459\) 15718.4 1875.70i 1.59842 0.190741i
\(460\) 0 0
\(461\) 3283.79 5687.69i 0.331760 0.574625i −0.651097 0.758995i \(-0.725691\pi\)
0.982857 + 0.184369i \(0.0590242\pi\)
\(462\) 0 0
\(463\) −4120.60 7137.09i −0.413608 0.716390i 0.581673 0.813423i \(-0.302398\pi\)
−0.995281 + 0.0970322i \(0.969065\pi\)
\(464\) 0 0
\(465\) −4478.97 2354.67i −0.446682 0.234828i
\(466\) 0 0
\(467\) −11330.9 −1.12276 −0.561382 0.827557i \(-0.689730\pi\)
−0.561382 + 0.827557i \(0.689730\pi\)
\(468\) 0 0
\(469\) −5567.93 12621.2i −0.548194 1.24263i
\(470\) 0 0
\(471\) −2535.30 + 1600.79i −0.248026 + 0.156604i
\(472\) 0 0
\(473\) −5488.35 + 3168.70i −0.533519 + 0.308027i
\(474\) 0 0
\(475\) −9946.74 5742.76i −0.960817 0.554728i
\(476\) 0 0
\(477\) 1170.26 + 14748.9i 0.112333 + 1.41574i
\(478\) 0 0
\(479\) 4151.12 7189.94i 0.395969 0.685839i −0.597255 0.802051i \(-0.703742\pi\)
0.993224 + 0.116212i \(0.0370753\pi\)
\(480\) 0 0
\(481\) 10056.4 5806.08i 0.953292 0.550384i
\(482\) 0 0
\(483\) −12427.0 + 6080.76i −1.17070 + 0.572845i
\(484\) 0 0
\(485\) 5789.46i 0.542033i
\(486\) 0 0
\(487\) −76.4693 −0.00711530 −0.00355765 0.999994i \(-0.501132\pi\)
−0.00355765 + 0.999994i \(0.501132\pi\)
\(488\) 0 0
\(489\) 8285.17 328.180i 0.766193 0.0303493i
\(490\) 0 0
\(491\) 6951.74 4013.59i 0.638957 0.368902i −0.145256 0.989394i \(-0.546401\pi\)
0.784213 + 0.620492i \(0.213067\pi\)
\(492\) 0 0
\(493\) 23345.9 + 13478.7i 2.13275 + 1.23134i
\(494\) 0 0
\(495\) −186.195 2346.63i −0.0169067 0.213077i
\(496\) 0 0
\(497\) −271.704 + 2503.09i −0.0245223 + 0.225913i
\(498\) 0 0
\(499\) 8035.68 + 13918.2i 0.720895 + 1.24863i 0.960641 + 0.277791i \(0.0896023\pi\)
−0.239747 + 0.970836i \(0.577064\pi\)
\(500\) 0 0
\(501\) −1025.98 1624.93i −0.0914918 0.144903i
\(502\) 0 0
\(503\) −7450.97 −0.660481 −0.330241 0.943897i \(-0.607130\pi\)
−0.330241 + 0.943897i \(0.607130\pi\)
\(504\) 0 0
\(505\) −882.452 −0.0777596
\(506\) 0 0
\(507\) 969.941 1844.99i 0.0849637 0.161615i
\(508\) 0 0
\(509\) −6060.49 10497.1i −0.527753 0.914095i −0.999477 0.0323486i \(-0.989701\pi\)
0.471724 0.881746i \(-0.343632\pi\)
\(510\) 0 0
\(511\) 933.918 8603.77i 0.0808495 0.744830i
\(512\) 0 0
\(513\) −1852.86 15527.1i −0.159466 1.33633i
\(514\) 0 0
\(515\) 3219.57 + 1858.82i 0.275478 + 0.159047i
\(516\) 0 0
\(517\) 3609.06 2083.69i 0.307014 0.177255i
\(518\) 0 0
\(519\) 4708.10 8955.57i 0.398194 0.757430i
\(520\) 0 0
\(521\) 10433.7 0.877371 0.438685 0.898641i \(-0.355444\pi\)
0.438685 + 0.898641i \(0.355444\pi\)
\(522\) 0 0
\(523\) 22346.3i 1.86833i 0.356846 + 0.934163i \(0.383852\pi\)
−0.356846 + 0.934163i \(0.616148\pi\)
\(524\) 0 0
\(525\) 5534.80 8228.36i 0.460111 0.684029i
\(526\) 0 0
\(527\) 20309.8 11725.9i 1.67876 0.969234i
\(528\) 0 0
\(529\) 4250.53 7362.13i 0.349349 0.605090i
\(530\) 0 0
\(531\) 1345.45 + 1954.96i 0.109958 + 0.159771i
\(532\) 0 0
\(533\) 11754.5 + 6786.45i 0.955240 + 0.551508i
\(534\) 0 0
\(535\) 5678.89 3278.71i 0.458916 0.264955i
\(536\) 0 0
\(537\) −175.207 4423.24i −0.0140796 0.355451i
\(538\) 0 0
\(539\) 4713.95 + 4302.94i 0.376706 + 0.343860i
\(540\) 0 0
\(541\) −13690.0 −1.08794 −0.543972 0.839103i \(-0.683080\pi\)
−0.543972 + 0.839103i \(0.683080\pi\)
\(542\) 0 0
\(543\) 131.471 + 3319.09i 0.0103903 + 0.262312i
\(544\) 0 0
\(545\) 5007.46 + 8673.18i 0.393571 + 0.681685i
\(546\) 0 0
\(547\) −8042.35 + 13929.8i −0.628640 + 1.08884i 0.359185 + 0.933266i \(0.383055\pi\)
−0.987825 + 0.155570i \(0.950279\pi\)
\(548\) 0 0
\(549\) 8793.49 + 4187.38i 0.683601 + 0.325524i
\(550\) 0 0
\(551\) 13314.6 23061.6i 1.02944 1.78305i
\(552\) 0 0
\(553\) 4203.50 5744.40i 0.323239 0.441730i
\(554\) 0 0
\(555\) 2961.12 + 4689.77i 0.226473 + 0.358684i
\(556\) 0 0
\(557\) 9207.28i 0.700404i 0.936674 + 0.350202i \(0.113887\pi\)
−0.936674 + 0.350202i \(0.886113\pi\)
\(558\) 0 0
\(559\) 17359.8i 1.31349i
\(560\) 0 0
\(561\) 9656.61 + 5076.65i 0.726743 + 0.382061i
\(562\) 0 0
\(563\) −10095.6 17486.0i −0.755732 1.30897i −0.945010 0.327042i \(-0.893948\pi\)
0.189278 0.981924i \(-0.439385\pi\)
\(564\) 0 0
\(565\) 7028.77 + 4058.06i 0.523367 + 0.302166i
\(566\) 0 0
\(567\) 13484.2 677.954i 0.998738 0.0502141i
\(568\) 0 0
\(569\) −3118.35 1800.38i −0.229750 0.132646i 0.380707 0.924696i \(-0.375681\pi\)
−0.610457 + 0.792050i \(0.709014\pi\)
\(570\) 0 0
\(571\) −1111.32 1924.86i −0.0814489 0.141074i 0.822424 0.568876i \(-0.192621\pi\)
−0.903873 + 0.427802i \(0.859288\pi\)
\(572\) 0 0
\(573\) 16570.4 + 8711.36i 1.20810 + 0.635117i
\(574\) 0 0
\(575\) 14814.5i 1.07444i
\(576\) 0 0
\(577\) 8818.52i 0.636256i −0.948048 0.318128i \(-0.896946\pi\)
0.948048 0.318128i \(-0.103054\pi\)
\(578\) 0 0
\(579\) −9681.82 15333.9i −0.694927 1.10061i
\(580\) 0 0
\(581\) −13063.4 + 17852.0i −0.932805 + 1.27475i
\(582\) 0 0
\(583\) −5098.33 + 8830.57i −0.362181 + 0.627316i
\(584\) 0 0
\(585\) 5821.86 + 2772.31i 0.411460 + 0.195933i
\(586\) 0 0
\(587\) −9540.65 + 16524.9i −0.670843 + 1.16193i 0.306823 + 0.951767i \(0.400734\pi\)
−0.977665 + 0.210167i \(0.932599\pi\)
\(588\) 0 0
\(589\) −11583.1 20062.5i −0.810311 1.40350i
\(590\) 0 0
\(591\) 392.783 + 9916.13i 0.0273383 + 0.690178i
\(592\) 0 0
\(593\) 7386.65 0.511523 0.255762 0.966740i \(-0.417674\pi\)
0.255762 + 0.966740i \(0.417674\pi\)
\(594\) 0 0
\(595\) −3951.88 8958.02i −0.272288 0.617215i
\(596\) 0 0
\(597\) −146.638 3701.98i −0.0100527 0.253789i
\(598\) 0 0
\(599\) 5442.40 3142.17i 0.371236 0.214333i −0.302762 0.953066i \(-0.597909\pi\)
0.673998 + 0.738733i \(0.264575\pi\)
\(600\) 0 0
\(601\) −23587.5 13618.2i −1.60092 0.924291i −0.991304 0.131594i \(-0.957991\pi\)
−0.609615 0.792697i \(-0.708676\pi\)
\(602\) 0 0
\(603\) −11401.6 16566.7i −0.769998 1.11882i
\(604\) 0 0
\(605\) −2306.95 + 3995.76i −0.155026 + 0.268514i
\(606\) 0 0
\(607\) 2294.69 1324.84i 0.153441 0.0885891i −0.421314 0.906915i \(-0.638431\pi\)
0.574755 + 0.818326i \(0.305098\pi\)
\(608\) 0 0
\(609\) 19077.5 + 12832.5i 1.26939 + 0.853857i
\(610\) 0 0
\(611\) 11415.5i 0.755848i
\(612\) 0 0
\(613\) 19866.5 1.30897 0.654486 0.756074i \(-0.272885\pi\)
0.654486 + 0.756074i \(0.272885\pi\)
\(614\) 0 0
\(615\) −3016.69 + 5738.24i −0.197796 + 0.376241i
\(616\) 0 0
\(617\) −21498.0 + 12411.9i −1.40272 + 0.809858i −0.994671 0.103104i \(-0.967123\pi\)
−0.408044 + 0.912962i \(0.633789\pi\)
\(618\) 0 0
\(619\) −6535.31 3773.16i −0.424356 0.245002i 0.272583 0.962132i \(-0.412122\pi\)
−0.696939 + 0.717130i \(0.745455\pi\)
\(620\) 0 0
\(621\) −16149.3 + 12083.4i −1.04356 + 0.780823i
\(622\) 0 0
\(623\) −25512.4 2769.31i −1.64066 0.178090i
\(624\) 0 0
\(625\) −3937.30 6819.61i −0.251987 0.436455i
\(626\) 0 0
\(627\) 5014.84 9539.05i 0.319415 0.607580i
\(628\) 0 0
\(629\) −25704.9 −1.62944
\(630\) 0 0
\(631\) −10950.1 −0.690837 −0.345418 0.938449i \(-0.612263\pi\)
−0.345418 + 0.938449i \(0.612263\pi\)
\(632\) 0 0
\(633\) 4375.37 + 6929.62i 0.274732 + 0.435115i
\(634\) 0 0
\(635\) 1805.02 + 3126.39i 0.112803 + 0.195381i
\(636\) 0 0
\(637\) −16664.1 + 5289.33i −1.03651 + 0.328996i
\(638\) 0 0
\(639\) 290.333 + 3659.10i 0.0179740 + 0.226528i
\(640\) 0 0
\(641\) 27715.4 + 16001.5i 1.70779 + 0.985994i 0.937287 + 0.348559i \(0.113329\pi\)
0.770504 + 0.637435i \(0.220004\pi\)
\(642\) 0 0
\(643\) 4359.15 2516.76i 0.267353 0.154356i −0.360331 0.932824i \(-0.617336\pi\)
0.627684 + 0.778468i \(0.284003\pi\)
\(644\) 0 0
\(645\) 8285.13 328.178i 0.505778 0.0200341i
\(646\) 0 0
\(647\) 5406.97 0.328547 0.164274 0.986415i \(-0.447472\pi\)
0.164274 + 0.986415i \(0.447472\pi\)
\(648\) 0 0
\(649\) 1635.58i 0.0989245i
\(650\) 0 0
\(651\) 17966.3 8791.20i 1.08165 0.529269i
\(652\) 0 0
\(653\) −22480.8 + 12979.3i −1.34723 + 0.777826i −0.987857 0.155367i \(-0.950344\pi\)
−0.359377 + 0.933193i \(0.617011\pi\)
\(654\) 0 0
\(655\) 2294.90 3974.89i 0.136900 0.237117i
\(656\) 0 0
\(657\) −997.950 12577.3i −0.0592599 0.746858i
\(658\) 0 0
\(659\) 22849.6 + 13192.2i 1.35067 + 0.779811i 0.988344 0.152240i \(-0.0486486\pi\)
0.362328 + 0.932051i \(0.381982\pi\)
\(660\) 0 0
\(661\) −4642.84 + 2680.54i −0.273200 + 0.157732i −0.630341 0.776318i \(-0.717085\pi\)
0.357141 + 0.934051i \(0.383752\pi\)
\(662\) 0 0
\(663\) −25269.2 + 15955.0i −1.48020 + 0.934599i
\(664\) 0 0
\(665\) −8848.96 + 3903.77i −0.516012 + 0.227642i
\(666\) 0 0
\(667\) −34347.5 −1.99391
\(668\) 0 0
\(669\) −2191.08 1151.89i −0.126625 0.0665688i
\(670\) 0 0
\(671\) 3356.18 + 5813.08i 0.193091 + 0.334443i
\(672\) 0 0
\(673\) 3378.73 5852.13i 0.193522 0.335190i −0.752893 0.658143i \(-0.771342\pi\)
0.946415 + 0.322953i \(0.104676\pi\)
\(674\) 0 0
\(675\) 5694.11 13288.5i 0.324691 0.757742i
\(676\) 0 0
\(677\) 2962.04 5130.41i 0.168154 0.291252i −0.769617 0.638506i \(-0.779553\pi\)
0.937771 + 0.347254i \(0.112886\pi\)
\(678\) 0 0
\(679\) −18468.0 13514.1i −1.04379 0.763804i
\(680\) 0 0
\(681\) 6706.63 12757.1i 0.377384 0.717847i
\(682\) 0 0
\(683\) 8838.52i 0.495163i 0.968867 + 0.247582i \(0.0796358\pi\)
−0.968867 + 0.247582i \(0.920364\pi\)
\(684\) 0 0
\(685\) 987.135i 0.0550606i
\(686\) 0 0
\(687\) 8617.21 5440.91i 0.478555 0.302160i
\(688\) 0 0
\(689\) −13965.7 24189.2i −0.772205 1.33750i
\(690\) 0 0
\(691\) 25392.1 + 14660.1i 1.39792 + 0.807087i 0.994174 0.107787i \(-0.0343763\pi\)
0.403741 + 0.914873i \(0.367710\pi\)
\(692\) 0 0
\(693\) 7920.22 + 4883.68i 0.434147 + 0.267700i
\(694\) 0 0
\(695\) −3422.75 1976.12i −0.186809 0.107854i
\(696\) 0 0
\(697\) −15022.6 26019.9i −0.816387 1.41402i
\(698\) 0 0
\(699\) −973.794 24584.2i −0.0526928 1.33027i
\(700\) 0 0
\(701\) 15890.7i 0.856184i −0.903735 0.428092i \(-0.859186\pi\)
0.903735 0.428092i \(-0.140814\pi\)
\(702\) 0 0
\(703\) 25391.9i 1.36227i
\(704\) 0 0
\(705\) −5448.18 + 215.805i −0.291050 + 0.0115287i
\(706\) 0 0
\(707\) 2059.87 2814.96i 0.109575 0.149742i
\(708\) 0 0
\(709\) 1060.77 1837.31i 0.0561890 0.0973222i −0.836563 0.547871i \(-0.815438\pi\)
0.892752 + 0.450549i \(0.148772\pi\)
\(710\) 0 0
\(711\) 4461.52 9369.19i 0.235331 0.494194i
\(712\) 0 0
\(713\) −14940.3 + 25877.4i −0.784740 + 1.35921i
\(714\) 0 0
\(715\) 2222.01 + 3848.63i 0.116222 + 0.201302i
\(716\) 0 0
\(717\) −5354.79 + 3381.02i −0.278910 + 0.176104i
\(718\) 0 0
\(719\) 23897.3 1.23953 0.619764 0.784789i \(-0.287228\pi\)
0.619764 + 0.784789i \(0.287228\pi\)
\(720\) 0 0
\(721\) −13444.8 + 5931.25i −0.694467 + 0.306368i
\(722\) 0 0
\(723\) 9478.57 + 4983.04i 0.487568 + 0.256323i
\(724\) 0 0
\(725\) 21321.2 12309.8i 1.09221 0.630586i
\(726\) 0 0
\(727\) 10755.1 + 6209.44i 0.548670 + 0.316775i 0.748586 0.663038i \(-0.230733\pi\)
−0.199915 + 0.979813i \(0.564067\pi\)
\(728\) 0 0
\(729\) 19130.3 4631.64i 0.971920 0.235311i
\(730\) 0 0
\(731\) −19213.9 + 33279.5i −0.972166 + 1.68384i
\(732\) 0 0
\(733\) −199.317 + 115.076i −0.0100436 + 0.00579865i −0.505013 0.863112i \(-0.668512\pi\)
0.494970 + 0.868910i \(0.335179\pi\)
\(734\) 0 0
\(735\) −2839.41 7853.12i −0.142494 0.394104i
\(736\) 0 0
\(737\) 13860.2i 0.692734i
\(738\) 0 0
\(739\) 6421.89 0.319666 0.159833 0.987144i \(-0.448904\pi\)
0.159833 + 0.987144i \(0.448904\pi\)
\(740\) 0 0
\(741\) 15760.7 + 24961.5i 0.781355 + 1.23750i
\(742\) 0 0
\(743\) −693.090 + 400.156i −0.0342221 + 0.0197581i −0.517013 0.855977i \(-0.672956\pi\)
0.482791 + 0.875735i \(0.339623\pi\)
\(744\) 0 0
\(745\) −5328.78 3076.58i −0.262056 0.151298i
\(746\) 0 0
\(747\) −13865.2 + 29117.0i −0.679119 + 1.42615i
\(748\) 0 0
\(749\) −2797.12 + 25768.6i −0.136455 + 1.25710i
\(750\) 0 0
\(751\) −971.585 1682.83i −0.0472086 0.0817676i 0.841456 0.540326i \(-0.181699\pi\)
−0.888664 + 0.458559i \(0.848366\pi\)
\(752\) 0 0
\(753\) 15401.2 610.050i 0.745353 0.0295238i
\(754\) 0 0
\(755\) 16293.5 0.785406
\(756\) 0 0
\(757\) 16016.6 0.768999 0.384500 0.923125i \(-0.374374\pi\)
0.384500 + 0.923125i \(0.374374\pi\)
\(758\) 0 0
\(759\) −13889.6 + 550.175i −0.664243 + 0.0263110i
\(760\) 0 0
\(761\) −20615.4 35706.8i −0.982005 1.70088i −0.654555 0.756014i \(-0.727144\pi\)
−0.327450 0.944869i \(-0.606189\pi\)
\(762\) 0 0
\(763\) −39355.5 4271.94i −1.86732 0.202693i
\(764\) 0 0
\(765\) −8092.37 11758.3i −0.382458 0.555717i
\(766\) 0 0
\(767\) −3880.03 2240.13i −0.182659 0.105458i
\(768\) 0 0
\(769\) 4147.70 2394.68i 0.194499 0.112294i −0.399588 0.916695i \(-0.630847\pi\)
0.594087 + 0.804401i \(0.297513\pi\)
\(770\) 0 0
\(771\) −6286.75 9956.83i −0.293660 0.465093i
\(772\) 0 0
\(773\) −12950.9 −0.602601 −0.301301 0.953529i \(-0.597421\pi\)
−0.301301 + 0.953529i \(0.597421\pi\)
\(774\) 0 0
\(775\) 21417.9i 0.992713i
\(776\) 0 0
\(777\) −21872.1 1501.34i −1.00985 0.0693181i
\(778\) 0 0
\(779\) −25703.1 + 14839.7i −1.18217 + 0.682526i
\(780\) 0 0
\(781\) −1264.86 + 2190.80i −0.0579515 + 0.100375i
\(782\) 0 0
\(783\) 30809.6 + 13201.8i 1.40619 + 0.602548i
\(784\) 0 0
\(785\) 2341.42 + 1351.82i 0.106457 + 0.0614632i
\(786\) 0 0
\(787\) −2678.82 + 1546.62i −0.121334 + 0.0700521i −0.559439 0.828872i \(-0.688983\pi\)
0.438105 + 0.898924i \(0.355650\pi\)
\(788\) 0 0
\(789\) −8391.96 4411.80i −0.378659 0.199067i
\(790\) 0 0
\(791\) −29351.9 + 12948.7i −1.31938 + 0.582053i
\(792\) 0 0
\(793\) −18386.9 −0.823377
\(794\) 0 0
\(795\) 11280.5 7122.54i 0.503245 0.317749i
\(796\) 0 0
\(797\) 5377.34 + 9313.83i 0.238990 + 0.413943i 0.960425 0.278539i \(-0.0898502\pi\)
−0.721434 + 0.692483i \(0.756517\pi\)
\(798\) 0 0
\(799\) 12634.8 21884.1i 0.559434 0.968968i
\(800\) 0 0
\(801\) −37294.8 + 2959.18i −1.64513 + 0.130534i
\(802\) 0 0
\(803\) 4347.64 7530.34i 0.191065 0.330934i
\(804\) 0 0
\(805\) 10067.5 + 7366.97i 0.440786 + 0.322548i
\(806\) 0 0
\(807\) −29163.4 + 1155.18i −1.27212 + 0.0503893i
\(808\) 0 0
\(809\) 29409.3i 1.27809i 0.769169 + 0.639045i \(0.220670\pi\)
−0.769169 + 0.639045i \(0.779330\pi\)
\(810\) 0 0
\(811\) 8850.91i 0.383228i −0.981470 0.191614i \(-0.938628\pi\)
0.981470 0.191614i \(-0.0613721\pi\)
\(812\) 0 0
\(813\) 1152.36 + 29092.4i 0.0497112 + 1.25500i
\(814\) 0 0
\(815\) −3738.31 6474.95i −0.160672 0.278291i
\(816\) 0 0
\(817\) 32874.4 + 18980.0i 1.40775 + 0.812763i
\(818\) 0 0
\(819\) −22433.2 + 12100.0i −0.957117 + 0.516251i
\(820\) 0 0
\(821\) −23750.3 13712.2i −1.00961 0.582899i −0.0985331 0.995134i \(-0.531415\pi\)
−0.911078 + 0.412235i \(0.864748\pi\)
\(822\) 0 0
\(823\) 12157.4 + 21057.2i 0.514921 + 0.891869i 0.999850 + 0.0173159i \(0.00551209\pi\)
−0.484929 + 0.874554i \(0.661155\pi\)
\(824\) 0 0
\(825\) 8424.79 5319.42i 0.355532 0.224483i
\(826\) 0 0
\(827\) 14415.0i 0.606119i −0.952972 0.303059i \(-0.901992\pi\)
0.952972 0.303059i \(-0.0980081\pi\)
\(828\) 0 0
\(829\) 8609.61i 0.360704i −0.983602 0.180352i \(-0.942276\pi\)
0.983602 0.180352i \(-0.0577238\pi\)
\(830\) 0 0
\(831\) 11264.4 21426.8i 0.470227 0.894449i
\(832\) 0 0
\(833\) 37800.2 + 8304.07i 1.57227 + 0.345401i
\(834\) 0 0
\(835\) −866.413 + 1500.67i −0.0359083 + 0.0621950i
\(836\) 0 0
\(837\) 23347.7 17469.5i 0.964176 0.721427i
\(838\) 0 0
\(839\) −3728.54 + 6458.02i −0.153425 + 0.265740i −0.932484 0.361210i \(-0.882364\pi\)
0.779059 + 0.626950i \(0.215697\pi\)
\(840\) 0 0
\(841\) 16345.9 + 28311.9i 0.670216 + 1.16085i
\(842\) 0 0
\(843\) 10302.2 + 5416.03i 0.420908 + 0.221279i
\(844\) 0 0
\(845\) −1879.52 −0.0765177
\(846\) 0 0
\(847\) −7361.18 16686.1i −0.298623 0.676910i
\(848\) 0 0
\(849\) −14000.0 + 8839.58i −0.565933 + 0.357330i
\(850\) 0 0
\(851\) 28363.6 16375.8i 1.14253 0.659640i
\(852\) 0 0
\(853\) −29109.8 16806.6i −1.16847 0.674615i −0.215148 0.976581i \(-0.569024\pi\)
−0.953318 + 0.301967i \(0.902357\pi\)
\(854\) 0 0
\(855\) −11615.2 + 7993.85i −0.464598 + 0.319747i
\(856\) 0 0
\(857\) 20047.3 34723.0i 0.799071 1.38403i −0.121151 0.992634i \(-0.538659\pi\)
0.920222 0.391397i \(-0.128008\pi\)
\(858\) 0 0
\(859\) −15246.8 + 8802.77i −0.605606 + 0.349647i −0.771244 0.636540i \(-0.780365\pi\)
0.165638 + 0.986187i \(0.447032\pi\)
\(860\) 0 0
\(861\) −11262.9 23017.5i −0.445804 0.911075i
\(862\) 0 0
\(863\) 36465.5i 1.43835i 0.694827 + 0.719177i \(0.255481\pi\)
−0.694827 + 0.719177i \(0.744519\pi\)
\(864\) 0 0
\(865\) −9123.19 −0.358610
\(866\) 0 0
\(867\) 40592.6 1607.89i 1.59008 0.0629838i
\(868\) 0 0
\(869\) 6193.65 3575.91i 0.241778 0.139591i
\(870\) 0 0
\(871\) 32880.0 + 18983.3i 1.27910 + 0.738489i
\(872\) 0 0
\(873\) −30121.6 14343.6i −1.16777 0.556079i
\(874\) 0 0
\(875\) −19673.1 2135.47i −0.760084 0.0825052i
\(876\) 0 0
\(877\) −16637.8 28817.5i −0.640614 1.10958i −0.985296 0.170856i \(-0.945347\pi\)
0.344682 0.938719i \(-0.387987\pi\)
\(878\) 0 0
\(879\) 25320.1 + 40101.4i 0.971586 + 1.53878i
\(880\) 0 0
\(881\) 11874.3 0.454091 0.227045 0.973884i \(-0.427093\pi\)
0.227045 + 0.973884i \(0.427093\pi\)
\(882\) 0 0
\(883\) −17571.9 −0.669697 −0.334848 0.942272i \(-0.608685\pi\)
−0.334848 + 0.942272i \(0.608685\pi\)
\(884\) 0 0
\(885\) 995.777 1894.13i 0.0378222 0.0719441i
\(886\) 0 0
\(887\) −14706.9 25473.1i −0.556719 0.964266i −0.997768 0.0667825i \(-0.978727\pi\)
0.441048 0.897483i \(-0.354607\pi\)
\(888\) 0 0
\(889\) −14186.3 1539.89i −0.535202 0.0580948i
\(890\) 0 0
\(891\) 12670.4 + 4845.12i 0.476403 + 0.182175i
\(892\) 0 0
\(893\) −21617.7 12481.0i −0.810089 0.467705i
\(894\) 0 0
\(895\) −3456.81 + 1995.79i −0.129104 + 0.0745385i
\(896\) 0 0
\(897\) 17718.4 33703.4i 0.659534 1.25454i
\(898\) 0 0
\(899\) 49657.5 1.84224
\(900\) 0 0
\(901\) 61829.3i 2.28616i
\(902\) 0 0
\(903\) −18292.7 + 27195.0i −0.674135 + 1.00221i
\(904\) 0 0
\(905\) 2593.90 1497.59i 0.0952753 0.0550072i
\(906\) 0 0
\(907\) 6950.77 12039.1i 0.254462 0.440740i −0.710288 0.703912i \(-0.751435\pi\)
0.964749 + 0.263171i \(0.0847684\pi\)
\(908\) 0 0
\(909\) 2186.30 4591.24i 0.0797746 0.167527i
\(910\) 0 0
\(911\) −40292.9 23263.1i −1.46538 0.846038i −0.466129 0.884717i \(-0.654352\pi\)
−0.999252 + 0.0386789i \(0.987685\pi\)
\(912\) 0 0
\(913\) −19248.2 + 11113.0i −0.697725 + 0.402832i
\(914\) 0 0
\(915\) −347.596 8775.34i −0.0125587 0.317053i
\(916\) 0 0
\(917\) 7322.74 + 16599.0i 0.263706 + 0.597761i
\(918\) 0 0
\(919\) 48733.4 1.74926 0.874628 0.484794i \(-0.161106\pi\)
0.874628 + 0.484794i \(0.161106\pi\)
\(920\) 0 0
\(921\) −975.776 24634.2i −0.0349109 0.881353i
\(922\) 0 0
\(923\) −3464.77 6001.16i −0.123558 0.214009i
\(924\) 0 0
\(925\) −11737.8 + 20330.5i −0.417229 + 0.722662i
\(926\) 0 0
\(927\) −17647.7 + 12145.6i −0.625272 + 0.430327i
\(928\) 0 0
\(929\) 24705.0 42790.3i 0.872491 1.51120i 0.0130789 0.999914i \(-0.495837\pi\)
0.859412 0.511284i \(-0.170830\pi\)
\(930\) 0 0
\(931\) 8202.97 37340.0i 0.288766 1.31447i
\(932\) 0 0
\(933\) 6045.56 + 9574.85i 0.212136 + 0.335977i
\(934\) 0 0
\(935\) 9837.35i 0.344081i
\(936\) 0 0
\(937\) 37596.2i 1.31080i −0.755284 0.655398i \(-0.772501\pi\)
0.755284 0.655398i \(-0.227499\pi\)
\(938\) 0 0
\(939\) 9309.07 + 4893.94i 0.323525 + 0.170083i
\(940\) 0 0
\(941\) 4437.16 + 7685.39i 0.153717 + 0.266245i 0.932591 0.360935i \(-0.117542\pi\)
−0.778874 + 0.627180i \(0.784209\pi\)
\(942\) 0 0
\(943\) 33152.9 + 19140.8i 1.14486 + 0.660987i
\(944\) 0 0
\(945\) −6198.96 10477.7i −0.213388 0.360677i
\(946\) 0 0
\(947\) 40026.5 + 23109.3i 1.37348 + 0.792979i 0.991364 0.131136i \(-0.0418623\pi\)
0.382115 + 0.924115i \(0.375196\pi\)
\(948\) 0 0
\(949\) 11909.3 + 20627.5i 0.407369 + 0.705583i
\(950\) 0 0
\(951\) 1008.78 + 530.332i 0.0343974 + 0.0180833i
\(952\) 0 0
\(953\) 23772.8i 0.808054i 0.914747 + 0.404027i \(0.132390\pi\)
−0.914747 + 0.404027i \(0.867610\pi\)
\(954\) 0 0
\(955\) 16880.6i 0.571981i
\(956\) 0 0
\(957\) 12333.1 + 19533.0i 0.416587 + 0.659782i
\(958\) 0 0
\(959\) −3148.89 2304.22i −0.106030 0.0775884i
\(960\) 0 0
\(961\) 6704.33 11612.2i 0.225046 0.389790i
\(962\) 0 0
\(963\) 2988.90 + 37669.4i 0.100017 + 1.26052i
\(964\) 0 0
\(965\) −8176.04 + 14161.3i −0.272742 + 0.472403i
\(966\) 0 0
\(967\) 15698.7 + 27190.9i 0.522064 + 0.904242i 0.999671 + 0.0256679i \(0.00817125\pi\)
−0.477606 + 0.878574i \(0.658495\pi\)
\(968\) 0 0
\(969\) −2586.43 65296.5i −0.0857463 2.16473i
\(970\) 0 0
\(971\) 42214.9 1.39520 0.697600 0.716487i \(-0.254251\pi\)
0.697600 + 0.716487i \(0.254251\pi\)
\(972\) 0 0
\(973\) 14293.2 6305.55i 0.470936 0.207756i
\(974\) 0 0
\(975\) 1080.24 + 27271.5i 0.0354824 + 0.895782i
\(976\) 0 0
\(977\) 39073.6 22559.2i 1.27950 0.738722i 0.302747 0.953071i \(-0.402096\pi\)
0.976757 + 0.214349i \(0.0687631\pi\)
\(978\) 0 0
\(979\) −22329.4 12891.9i −0.728959 0.420865i
\(980\) 0 0
\(981\) −57531.2 + 4564.84i −1.87240 + 0.148567i
\(982\) 0 0
\(983\) −17928.1 + 31052.5i −0.581708 + 1.00755i 0.413569 + 0.910473i \(0.364282\pi\)
−0.995277 + 0.0970751i \(0.969051\pi\)
\(984\) 0 0
\(985\) 7749.56 4474.21i 0.250682 0.144731i
\(986\) 0 0
\(987\) 12029.0 17883.1i 0.387932 0.576722i
\(988\) 0 0
\(989\) 48962.3i 1.57423i
\(990\) 0 0
\(991\) 41167.8 1.31961 0.659807 0.751435i \(-0.270638\pi\)
0.659807 + 0.751435i \(0.270638\pi\)
\(992\) 0 0
\(993\) 9112.52 17333.5i 0.291216 0.553940i
\(994\) 0 0
\(995\) −2893.14 + 1670.35i −0.0921795 + 0.0532199i
\(996\) 0 0
\(997\) −30797.6 17781.0i −0.978304 0.564824i −0.0765462 0.997066i \(-0.524389\pi\)
−0.901758 + 0.432242i \(0.857723\pi\)
\(998\) 0 0
\(999\) −31736.3 + 3787.13i −1.00510 + 0.119940i
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 252.4.x.a.209.24 yes 48
3.2 odd 2 756.4.x.a.629.15 48
7.6 odd 2 inner 252.4.x.a.209.1 yes 48
9.2 odd 6 2268.4.f.a.1133.19 48
9.4 even 3 756.4.x.a.125.10 48
9.5 odd 6 inner 252.4.x.a.41.1 48
9.7 even 3 2268.4.f.a.1133.30 48
21.20 even 2 756.4.x.a.629.10 48
63.13 odd 6 756.4.x.a.125.15 48
63.20 even 6 2268.4.f.a.1133.29 48
63.34 odd 6 2268.4.f.a.1133.20 48
63.41 even 6 inner 252.4.x.a.41.24 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.x.a.41.1 48 9.5 odd 6 inner
252.4.x.a.41.24 yes 48 63.41 even 6 inner
252.4.x.a.209.1 yes 48 7.6 odd 2 inner
252.4.x.a.209.24 yes 48 1.1 even 1 trivial
756.4.x.a.125.10 48 9.4 even 3
756.4.x.a.125.15 48 63.13 odd 6
756.4.x.a.629.10 48 21.20 even 2
756.4.x.a.629.15 48 3.2 odd 2
2268.4.f.a.1133.19 48 9.2 odd 6
2268.4.f.a.1133.20 48 63.34 odd 6
2268.4.f.a.1133.29 48 63.20 even 6
2268.4.f.a.1133.30 48 9.7 even 3