Properties

Label 76.4.e.a.49.1
Level $76$
Weight $4$
Character 76.49
Analytic conductor $4.484$
Analytic rank $0$
Dimension $10$
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] = [76,4,Mod(45,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.45");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.e (of order \(3\), 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: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 90 x^{8} - 212 x^{7} + 7012 x^{6} - 14448 x^{5} + 100896 x^{4} - 25920 x^{3} + 783360 x^{2} - 774144 x + 1016064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.1
Root \(-4.43560 - 7.68268i\) of defining polynomial
Character \(\chi\) \(=\) 76.49
Dual form 76.4.e.a.45.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.93560 + 6.81665i) q^{3} +(0.0670670 - 0.116164i) q^{5} -9.41300 q^{7} +(-17.4778 - 30.2725i) q^{9} +O(q^{10})\) \(q+(-3.93560 + 6.81665i) q^{3} +(0.0670670 - 0.116164i) q^{5} -9.41300 q^{7} +(-17.4778 - 30.2725i) q^{9} -47.5639 q^{11} +(-22.2514 - 38.5406i) q^{13} +(0.527898 + 0.914345i) q^{15} +(-0.963494 + 1.66882i) q^{17} +(-42.7995 + 70.9028i) q^{19} +(37.0458 - 64.1651i) q^{21} +(77.1861 + 133.690i) q^{23} +(62.4910 + 108.238i) q^{25} +62.6206 q^{27} +(-26.5445 - 45.9764i) q^{29} +11.3029 q^{31} +(187.192 - 324.227i) q^{33} +(-0.631302 + 1.09345i) q^{35} -148.199 q^{37} +350.290 q^{39} +(-131.426 + 227.637i) q^{41} +(114.553 - 198.412i) q^{43} -4.68875 q^{45} +(176.617 + 305.910i) q^{47} -254.396 q^{49} +(-7.58385 - 13.1356i) q^{51} +(110.311 + 191.065i) q^{53} +(-3.18997 + 5.52519i) q^{55} +(-314.878 - 570.794i) q^{57} +(-421.722 + 730.444i) q^{59} +(74.9309 + 129.784i) q^{61} +(164.519 + 284.955i) q^{63} -5.96934 q^{65} +(-459.082 - 795.154i) q^{67} -1215.09 q^{69} +(493.025 - 853.944i) q^{71} +(461.416 - 799.196i) q^{73} -983.757 q^{75} +447.719 q^{77} +(-540.151 + 935.568i) q^{79} +(225.452 - 390.494i) q^{81} -316.723 q^{83} +(0.129237 + 0.223846i) q^{85} +417.873 q^{87} +(508.710 + 881.112i) q^{89} +(209.452 + 362.782i) q^{91} +(-44.4837 + 77.0481i) q^{93} +(5.36588 + 9.72698i) q^{95} +(537.224 - 930.499i) q^{97} +(831.315 + 1439.88i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 7 q^{3} - 4 q^{5} - 20 q^{7} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 7 q^{3} - 4 q^{5} - 20 q^{7} - 50 q^{9} - 50 q^{11} - 56 q^{13} + 10 q^{15} + 32 q^{17} + 77 q^{19} + 126 q^{21} + 184 q^{23} - 121 q^{25} - 218 q^{27} + 352 q^{29} + 264 q^{31} + 83 q^{33} - 132 q^{35} - 640 q^{37} - 324 q^{39} - 57 q^{41} - 528 q^{43} - 232 q^{45} - 434 q^{47} + 2138 q^{49} - 242 q^{51} + 780 q^{53} + 598 q^{55} + 1482 q^{57} - 343 q^{59} + 536 q^{61} - 1568 q^{63} - 1988 q^{65} + 779 q^{67} - 1156 q^{69} + 474 q^{71} + 1453 q^{73} - 2994 q^{75} - 2956 q^{77} - 1968 q^{79} - 1097 q^{81} - 698 q^{83} - 2334 q^{85} + 8372 q^{87} + 380 q^{89} + 1348 q^{91} + 1684 q^{93} + 4312 q^{95} + 883 q^{97} + 5230 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −3.93560 + 6.81665i −0.757406 + 1.31187i 0.186763 + 0.982405i \(0.440200\pi\)
−0.944169 + 0.329461i \(0.893133\pi\)
\(4\) 0 0
\(5\) 0.0670670 0.116164i 0.00599866 0.0103900i −0.863010 0.505186i \(-0.831424\pi\)
0.869009 + 0.494796i \(0.164757\pi\)
\(6\) 0 0
\(7\) −9.41300 −0.508254 −0.254127 0.967171i \(-0.581788\pi\)
−0.254127 + 0.967171i \(0.581788\pi\)
\(8\) 0 0
\(9\) −17.4778 30.2725i −0.647327 1.12120i
\(10\) 0 0
\(11\) −47.5639 −1.30373 −0.651867 0.758334i \(-0.726014\pi\)
−0.651867 + 0.758334i \(0.726014\pi\)
\(12\) 0 0
\(13\) −22.2514 38.5406i −0.474725 0.822248i 0.524856 0.851191i \(-0.324119\pi\)
−0.999581 + 0.0289429i \(0.990786\pi\)
\(14\) 0 0
\(15\) 0.527898 + 0.914345i 0.00908684 + 0.0157389i
\(16\) 0 0
\(17\) −0.963494 + 1.66882i −0.0137460 + 0.0238087i −0.872817 0.488048i \(-0.837709\pi\)
0.859071 + 0.511857i \(0.171042\pi\)
\(18\) 0 0
\(19\) −42.7995 + 70.9028i −0.516783 + 0.856116i
\(20\) 0 0
\(21\) 37.0458 64.1651i 0.384955 0.666761i
\(22\) 0 0
\(23\) 77.1861 + 133.690i 0.699757 + 1.21201i 0.968551 + 0.248817i \(0.0800417\pi\)
−0.268794 + 0.963198i \(0.586625\pi\)
\(24\) 0 0
\(25\) 62.4910 + 108.238i 0.499928 + 0.865901i
\(26\) 0 0
\(27\) 62.6206 0.446346
\(28\) 0 0
\(29\) −26.5445 45.9764i −0.169972 0.294400i 0.768438 0.639924i \(-0.221034\pi\)
−0.938410 + 0.345525i \(0.887701\pi\)
\(30\) 0 0
\(31\) 11.3029 0.0654859 0.0327430 0.999464i \(-0.489576\pi\)
0.0327430 + 0.999464i \(0.489576\pi\)
\(32\) 0 0
\(33\) 187.192 324.227i 0.987455 1.71032i
\(34\) 0 0
\(35\) −0.631302 + 1.09345i −0.00304884 + 0.00528075i
\(36\) 0 0
\(37\) −148.199 −0.658480 −0.329240 0.944246i \(-0.606792\pi\)
−0.329240 + 0.944246i \(0.606792\pi\)
\(38\) 0 0
\(39\) 350.290 1.43824
\(40\) 0 0
\(41\) −131.426 + 227.637i −0.500617 + 0.867094i 0.499383 + 0.866382i \(0.333560\pi\)
−1.00000 0.000712775i \(0.999773\pi\)
\(42\) 0 0
\(43\) 114.553 198.412i 0.406261 0.703665i −0.588206 0.808711i \(-0.700166\pi\)
0.994467 + 0.105046i \(0.0334990\pi\)
\(44\) 0 0
\(45\) −4.68875 −0.0155324
\(46\) 0 0
\(47\) 176.617 + 305.910i 0.548134 + 0.949395i 0.998402 + 0.0565023i \(0.0179948\pi\)
−0.450269 + 0.892893i \(0.648672\pi\)
\(48\) 0 0
\(49\) −254.396 −0.741678
\(50\) 0 0
\(51\) −7.58385 13.1356i −0.0208226 0.0360657i
\(52\) 0 0
\(53\) 110.311 + 191.065i 0.285894 + 0.495184i 0.972826 0.231539i \(-0.0743760\pi\)
−0.686931 + 0.726722i \(0.741043\pi\)
\(54\) 0 0
\(55\) −3.18997 + 5.52519i −0.00782065 + 0.0135458i
\(56\) 0 0
\(57\) −314.878 570.794i −0.731695 1.32638i
\(58\) 0 0
\(59\) −421.722 + 730.444i −0.930569 + 1.61179i −0.148218 + 0.988955i \(0.547354\pi\)
−0.782351 + 0.622838i \(0.785980\pi\)
\(60\) 0 0
\(61\) 74.9309 + 129.784i 0.157277 + 0.272412i 0.933886 0.357571i \(-0.116395\pi\)
−0.776609 + 0.629983i \(0.783062\pi\)
\(62\) 0 0
\(63\) 164.519 + 284.955i 0.329007 + 0.569856i
\(64\) 0 0
\(65\) −5.96934 −0.0113909
\(66\) 0 0
\(67\) −459.082 795.154i −0.837102 1.44990i −0.892307 0.451429i \(-0.850915\pi\)
0.0552050 0.998475i \(-0.482419\pi\)
\(68\) 0 0
\(69\) −1215.09 −2.12000
\(70\) 0 0
\(71\) 493.025 853.944i 0.824102 1.42739i −0.0785015 0.996914i \(-0.525014\pi\)
0.902604 0.430473i \(-0.141653\pi\)
\(72\) 0 0
\(73\) 461.416 799.196i 0.739790 1.28135i −0.212800 0.977096i \(-0.568258\pi\)
0.952590 0.304257i \(-0.0984083\pi\)
\(74\) 0 0
\(75\) −983.757 −1.51459
\(76\) 0 0
\(77\) 447.719 0.662627
\(78\) 0 0
\(79\) −540.151 + 935.568i −0.769262 + 1.33240i 0.168702 + 0.985667i \(0.446042\pi\)
−0.937964 + 0.346733i \(0.887291\pi\)
\(80\) 0 0
\(81\) 225.452 390.494i 0.309262 0.535658i
\(82\) 0 0
\(83\) −316.723 −0.418853 −0.209427 0.977824i \(-0.567160\pi\)
−0.209427 + 0.977824i \(0.567160\pi\)
\(84\) 0 0
\(85\) 0.129237 + 0.223846i 0.000164915 + 0.000285641i
\(86\) 0 0
\(87\) 417.873 0.514951
\(88\) 0 0
\(89\) 508.710 + 881.112i 0.605879 + 1.04941i 0.991912 + 0.126928i \(0.0405117\pi\)
−0.386033 + 0.922485i \(0.626155\pi\)
\(90\) 0 0
\(91\) 209.452 + 362.782i 0.241281 + 0.417911i
\(92\) 0 0
\(93\) −44.4837 + 77.0481i −0.0495994 + 0.0859087i
\(94\) 0 0
\(95\) 5.36588 + 9.72698i 0.00579503 + 0.0105049i
\(96\) 0 0
\(97\) 537.224 930.499i 0.562338 0.973998i −0.434954 0.900453i \(-0.643235\pi\)
0.997292 0.0735455i \(-0.0234314\pi\)
\(98\) 0 0
\(99\) 831.315 + 1439.88i 0.843942 + 1.46175i
\(100\) 0 0
\(101\) 642.035 + 1112.04i 0.632523 + 1.09556i 0.987034 + 0.160510i \(0.0513140\pi\)
−0.354511 + 0.935052i \(0.615353\pi\)
\(102\) 0 0
\(103\) 491.614 0.470293 0.235147 0.971960i \(-0.424443\pi\)
0.235147 + 0.971960i \(0.424443\pi\)
\(104\) 0 0
\(105\) −4.96910 8.60673i −0.00461842 0.00799934i
\(106\) 0 0
\(107\) 302.381 0.273199 0.136600 0.990626i \(-0.456383\pi\)
0.136600 + 0.990626i \(0.456383\pi\)
\(108\) 0 0
\(109\) 392.757 680.274i 0.345131 0.597784i −0.640247 0.768169i \(-0.721168\pi\)
0.985378 + 0.170385i \(0.0545012\pi\)
\(110\) 0 0
\(111\) 583.251 1010.22i 0.498736 0.863837i
\(112\) 0 0
\(113\) −171.809 −0.143030 −0.0715150 0.997440i \(-0.522783\pi\)
−0.0715150 + 0.997440i \(0.522783\pi\)
\(114\) 0 0
\(115\) 20.7066 0.0167904
\(116\) 0 0
\(117\) −777.813 + 1347.21i −0.614605 + 1.06453i
\(118\) 0 0
\(119\) 9.06936 15.7086i 0.00698645 0.0121009i
\(120\) 0 0
\(121\) 931.327 0.699720
\(122\) 0 0
\(123\) −1034.48 1791.77i −0.758341 1.31348i
\(124\) 0 0
\(125\) 33.5311 0.0239929
\(126\) 0 0
\(127\) −1367.41 2368.43i −0.955420 1.65484i −0.733405 0.679792i \(-0.762070\pi\)
−0.222015 0.975043i \(-0.571263\pi\)
\(128\) 0 0
\(129\) 901.672 + 1561.74i 0.615409 + 1.06592i
\(130\) 0 0
\(131\) 648.315 1122.91i 0.432393 0.748927i −0.564685 0.825306i \(-0.691003\pi\)
0.997079 + 0.0763788i \(0.0243358\pi\)
\(132\) 0 0
\(133\) 402.871 667.408i 0.262657 0.435125i
\(134\) 0 0
\(135\) 4.19978 7.27423i 0.00267748 0.00463753i
\(136\) 0 0
\(137\) 1042.84 + 1806.25i 0.650334 + 1.12641i 0.983042 + 0.183382i \(0.0587044\pi\)
−0.332708 + 0.943030i \(0.607962\pi\)
\(138\) 0 0
\(139\) −234.329 405.869i −0.142989 0.247665i 0.785632 0.618694i \(-0.212338\pi\)
−0.928621 + 0.371030i \(0.879005\pi\)
\(140\) 0 0
\(141\) −2780.38 −1.66064
\(142\) 0 0
\(143\) 1058.36 + 1833.14i 0.618915 + 1.07199i
\(144\) 0 0
\(145\) −7.12103 −0.00407841
\(146\) 0 0
\(147\) 1001.20 1734.13i 0.561751 0.972982i
\(148\) 0 0
\(149\) −1211.56 + 2098.49i −0.666141 + 1.15379i 0.312834 + 0.949808i \(0.398722\pi\)
−0.978975 + 0.203982i \(0.934612\pi\)
\(150\) 0 0
\(151\) −3484.73 −1.87804 −0.939018 0.343867i \(-0.888263\pi\)
−0.939018 + 0.343867i \(0.888263\pi\)
\(152\) 0 0
\(153\) 67.3592 0.0355926
\(154\) 0 0
\(155\) 0.758053 1.31299i 0.000392828 0.000680397i
\(156\) 0 0
\(157\) −1665.86 + 2885.36i −0.846817 + 1.46673i 0.0372162 + 0.999307i \(0.488151\pi\)
−0.884034 + 0.467423i \(0.845182\pi\)
\(158\) 0 0
\(159\) −1736.56 −0.866152
\(160\) 0 0
\(161\) −726.552 1258.43i −0.355654 0.616011i
\(162\) 0 0
\(163\) 2610.26 1.25430 0.627151 0.778897i \(-0.284221\pi\)
0.627151 + 0.778897i \(0.284221\pi\)
\(164\) 0 0
\(165\) −25.1089 43.4899i −0.0118468 0.0205193i
\(166\) 0 0
\(167\) 97.3470 + 168.610i 0.0451074 + 0.0781284i 0.887698 0.460427i \(-0.152304\pi\)
−0.842590 + 0.538555i \(0.818970\pi\)
\(168\) 0 0
\(169\) 108.250 187.495i 0.0492718 0.0853413i
\(170\) 0 0
\(171\) 2894.45 + 56.4205i 1.29441 + 0.0252315i
\(172\) 0 0
\(173\) 292.376 506.410i 0.128491 0.222553i −0.794601 0.607132i \(-0.792320\pi\)
0.923092 + 0.384579i \(0.125653\pi\)
\(174\) 0 0
\(175\) −588.228 1018.84i −0.254090 0.440098i
\(176\) 0 0
\(177\) −3319.46 5749.47i −1.40964 2.44156i
\(178\) 0 0
\(179\) −1465.40 −0.611894 −0.305947 0.952048i \(-0.598973\pi\)
−0.305947 + 0.952048i \(0.598973\pi\)
\(180\) 0 0
\(181\) 1179.10 + 2042.26i 0.484210 + 0.838676i 0.999835 0.0181381i \(-0.00577387\pi\)
−0.515626 + 0.856814i \(0.672441\pi\)
\(182\) 0 0
\(183\) −1179.59 −0.476491
\(184\) 0 0
\(185\) −9.93926 + 17.2153i −0.00394999 + 0.00684159i
\(186\) 0 0
\(187\) 45.8276 79.3756i 0.0179211 0.0310402i
\(188\) 0 0
\(189\) −589.448 −0.226857
\(190\) 0 0
\(191\) −4343.02 −1.64529 −0.822643 0.568558i \(-0.807501\pi\)
−0.822643 + 0.568558i \(0.807501\pi\)
\(192\) 0 0
\(193\) 401.381 695.212i 0.149700 0.259287i −0.781417 0.624009i \(-0.785503\pi\)
0.931116 + 0.364722i \(0.118836\pi\)
\(194\) 0 0
\(195\) 23.4929 40.6909i 0.00862750 0.0149433i
\(196\) 0 0
\(197\) −160.829 −0.0581654 −0.0290827 0.999577i \(-0.509259\pi\)
−0.0290827 + 0.999577i \(0.509259\pi\)
\(198\) 0 0
\(199\) 709.541 + 1228.96i 0.252754 + 0.437782i 0.964283 0.264874i \(-0.0853304\pi\)
−0.711529 + 0.702657i \(0.751997\pi\)
\(200\) 0 0
\(201\) 7227.05 2.53610
\(202\) 0 0
\(203\) 249.863 + 432.775i 0.0863888 + 0.149630i
\(204\) 0 0
\(205\) 17.6287 + 30.5338i 0.00600606 + 0.0104028i
\(206\) 0 0
\(207\) 2698.09 4673.23i 0.905944 1.56914i
\(208\) 0 0
\(209\) 2035.71 3372.41i 0.673747 1.11615i
\(210\) 0 0
\(211\) −1451.84 + 2514.65i −0.473689 + 0.820454i −0.999546 0.0301189i \(-0.990411\pi\)
0.525857 + 0.850573i \(0.323745\pi\)
\(212\) 0 0
\(213\) 3880.69 + 6721.55i 1.24836 + 2.16222i
\(214\) 0 0
\(215\) −15.3655 26.6138i −0.00487404 0.00844209i
\(216\) 0 0
\(217\) −106.394 −0.0332835
\(218\) 0 0
\(219\) 3631.89 + 6290.62i 1.12064 + 1.94101i
\(220\) 0 0
\(221\) 85.7564 0.0261022
\(222\) 0 0
\(223\) −1786.34 + 3094.04i −0.536423 + 0.929112i 0.462670 + 0.886531i \(0.346892\pi\)
−0.999093 + 0.0425814i \(0.986442\pi\)
\(224\) 0 0
\(225\) 2184.42 3783.52i 0.647234 1.12104i
\(226\) 0 0
\(227\) 193.355 0.0565349 0.0282674 0.999600i \(-0.491001\pi\)
0.0282674 + 0.999600i \(0.491001\pi\)
\(228\) 0 0
\(229\) 830.109 0.239542 0.119771 0.992802i \(-0.461784\pi\)
0.119771 + 0.992802i \(0.461784\pi\)
\(230\) 0 0
\(231\) −1762.04 + 3051.95i −0.501878 + 0.869278i
\(232\) 0 0
\(233\) −2541.17 + 4401.43i −0.714496 + 1.23754i 0.248658 + 0.968591i \(0.420011\pi\)
−0.963154 + 0.268952i \(0.913323\pi\)
\(234\) 0 0
\(235\) 47.3808 0.0131523
\(236\) 0 0
\(237\) −4251.63 7364.04i −1.16529 2.01834i
\(238\) 0 0
\(239\) 6668.03 1.80468 0.902341 0.431023i \(-0.141847\pi\)
0.902341 + 0.431023i \(0.141847\pi\)
\(240\) 0 0
\(241\) 89.9899 + 155.867i 0.0240529 + 0.0416609i 0.877801 0.479025i \(-0.159010\pi\)
−0.853748 + 0.520686i \(0.825676\pi\)
\(242\) 0 0
\(243\) 2619.95 + 4537.90i 0.691647 + 1.19797i
\(244\) 0 0
\(245\) −17.0616 + 29.5515i −0.00444907 + 0.00770602i
\(246\) 0 0
\(247\) 3684.98 + 71.8301i 0.949270 + 0.0185038i
\(248\) 0 0
\(249\) 1246.49 2158.99i 0.317242 0.549479i
\(250\) 0 0
\(251\) −2513.33 4353.22i −0.632032 1.09471i −0.987136 0.159885i \(-0.948888\pi\)
0.355103 0.934827i \(-0.384446\pi\)
\(252\) 0 0
\(253\) −3671.27 6358.83i −0.912296 1.58014i
\(254\) 0 0
\(255\) −2.03450 −0.000499630
\(256\) 0 0
\(257\) −1871.36 3241.29i −0.454210 0.786715i 0.544432 0.838805i \(-0.316745\pi\)
−0.998642 + 0.0520895i \(0.983412\pi\)
\(258\) 0 0
\(259\) 1395.00 0.334675
\(260\) 0 0
\(261\) −927.879 + 1607.13i −0.220055 + 0.381146i
\(262\) 0 0
\(263\) 125.709 217.734i 0.0294735 0.0510497i −0.850912 0.525308i \(-0.823950\pi\)
0.880386 + 0.474258i \(0.157284\pi\)
\(264\) 0 0
\(265\) 29.5930 0.00685993
\(266\) 0 0
\(267\) −8008.32 −1.83558
\(268\) 0 0
\(269\) −1085.34 + 1879.87i −0.246002 + 0.426088i −0.962413 0.271591i \(-0.912450\pi\)
0.716411 + 0.697679i \(0.245784\pi\)
\(270\) 0 0
\(271\) −670.094 + 1160.64i −0.150204 + 0.260161i −0.931302 0.364247i \(-0.881326\pi\)
0.781098 + 0.624408i \(0.214660\pi\)
\(272\) 0 0
\(273\) −3297.28 −0.730991
\(274\) 0 0
\(275\) −2972.32 5148.20i −0.651773 1.12890i
\(276\) 0 0
\(277\) 191.769 0.0415967 0.0207984 0.999784i \(-0.493379\pi\)
0.0207984 + 0.999784i \(0.493379\pi\)
\(278\) 0 0
\(279\) −197.551 342.168i −0.0423908 0.0734231i
\(280\) 0 0
\(281\) −142.001 245.953i −0.0301461 0.0522146i 0.850559 0.525880i \(-0.176264\pi\)
−0.880705 + 0.473665i \(0.842931\pi\)
\(282\) 0 0
\(283\) −44.9188 + 77.8016i −0.00943514 + 0.0163421i −0.870704 0.491807i \(-0.836337\pi\)
0.861269 + 0.508149i \(0.169670\pi\)
\(284\) 0 0
\(285\) −87.4234 1.70411i −0.0181702 0.000354186i
\(286\) 0 0
\(287\) 1237.11 2142.74i 0.254441 0.440704i
\(288\) 0 0
\(289\) 2454.64 + 4251.57i 0.499622 + 0.865371i
\(290\) 0 0
\(291\) 4228.59 + 7324.14i 0.851837 + 1.47542i
\(292\) 0 0
\(293\) −5957.69 −1.18789 −0.593946 0.804505i \(-0.702431\pi\)
−0.593946 + 0.804505i \(0.702431\pi\)
\(294\) 0 0
\(295\) 56.5673 + 97.9775i 0.0111643 + 0.0193372i
\(296\) 0 0
\(297\) −2978.48 −0.581916
\(298\) 0 0
\(299\) 3435.00 5949.59i 0.664385 1.15075i
\(300\) 0 0
\(301\) −1078.29 + 1867.65i −0.206484 + 0.357641i
\(302\) 0 0
\(303\) −10107.2 −1.91631
\(304\) 0 0
\(305\) 20.1016 0.00377381
\(306\) 0 0
\(307\) −1127.36 + 1952.64i −0.209582 + 0.363006i −0.951583 0.307392i \(-0.900544\pi\)
0.742001 + 0.670399i \(0.233877\pi\)
\(308\) 0 0
\(309\) −1934.80 + 3351.16i −0.356203 + 0.616961i
\(310\) 0 0
\(311\) 5929.03 1.08104 0.540521 0.841330i \(-0.318227\pi\)
0.540521 + 0.841330i \(0.318227\pi\)
\(312\) 0 0
\(313\) 3020.55 + 5231.75i 0.545469 + 0.944780i 0.998577 + 0.0533246i \(0.0169818\pi\)
−0.453108 + 0.891456i \(0.649685\pi\)
\(314\) 0 0
\(315\) 44.1352 0.00789439
\(316\) 0 0
\(317\) −3084.14 5341.88i −0.546443 0.946467i −0.998515 0.0544850i \(-0.982648\pi\)
0.452072 0.891982i \(-0.350685\pi\)
\(318\) 0 0
\(319\) 1262.56 + 2186.82i 0.221598 + 0.383819i
\(320\) 0 0
\(321\) −1190.05 + 2061.23i −0.206923 + 0.358401i
\(322\) 0 0
\(323\) −77.0870 139.739i −0.0132794 0.0240721i
\(324\) 0 0
\(325\) 2781.02 4816.88i 0.474657 0.822130i
\(326\) 0 0
\(327\) 3091.46 + 5354.57i 0.522808 + 0.905530i
\(328\) 0 0
\(329\) −1662.50 2879.53i −0.278591 0.482534i
\(330\) 0 0
\(331\) −6863.00 −1.13965 −0.569826 0.821765i \(-0.692989\pi\)
−0.569826 + 0.821765i \(0.692989\pi\)
\(332\) 0 0
\(333\) 2590.20 + 4486.35i 0.426252 + 0.738290i
\(334\) 0 0
\(335\) −123.157 −0.0200860
\(336\) 0 0
\(337\) 3914.15 6779.50i 0.632692 1.09585i −0.354308 0.935129i \(-0.615283\pi\)
0.986999 0.160725i \(-0.0513833\pi\)
\(338\) 0 0
\(339\) 676.169 1171.16i 0.108332 0.187636i
\(340\) 0 0
\(341\) −537.611 −0.0853762
\(342\) 0 0
\(343\) 5623.28 0.885215
\(344\) 0 0
\(345\) −81.4927 + 141.149i −0.0127172 + 0.0220268i
\(346\) 0 0
\(347\) 5492.82 9513.84i 0.849769 1.47184i −0.0316451 0.999499i \(-0.510075\pi\)
0.881414 0.472344i \(-0.156592\pi\)
\(348\) 0 0
\(349\) 2768.56 0.424635 0.212318 0.977201i \(-0.431899\pi\)
0.212318 + 0.977201i \(0.431899\pi\)
\(350\) 0 0
\(351\) −1393.40 2413.43i −0.211892 0.367007i
\(352\) 0 0
\(353\) −7098.29 −1.07027 −0.535133 0.844768i \(-0.679739\pi\)
−0.535133 + 0.844768i \(0.679739\pi\)
\(354\) 0 0
\(355\) −66.1314 114.543i −0.00988701 0.0171248i
\(356\) 0 0
\(357\) 71.3867 + 123.645i 0.0105832 + 0.0183306i
\(358\) 0 0
\(359\) 3319.66 5749.82i 0.488036 0.845304i −0.511869 0.859064i \(-0.671047\pi\)
0.999905 + 0.0137598i \(0.00438002\pi\)
\(360\) 0 0
\(361\) −3195.41 6069.21i −0.465871 0.884853i
\(362\) 0 0
\(363\) −3665.33 + 6348.53i −0.529972 + 0.917938i
\(364\) 0 0
\(365\) −61.8916 107.199i −0.00887549 0.0153728i
\(366\) 0 0
\(367\) 5043.80 + 8736.12i 0.717396 + 1.24257i 0.962028 + 0.272950i \(0.0879993\pi\)
−0.244633 + 0.969616i \(0.578667\pi\)
\(368\) 0 0
\(369\) 9188.17 1.29625
\(370\) 0 0
\(371\) −1038.36 1798.49i −0.145307 0.251679i
\(372\) 0 0
\(373\) 3174.01 0.440600 0.220300 0.975432i \(-0.429296\pi\)
0.220300 + 0.975432i \(0.429296\pi\)
\(374\) 0 0
\(375\) −131.965 + 228.570i −0.0181724 + 0.0314755i
\(376\) 0 0
\(377\) −1181.30 + 2046.08i −0.161380 + 0.279518i
\(378\) 0 0
\(379\) −2129.14 −0.288567 −0.144283 0.989536i \(-0.546088\pi\)
−0.144283 + 0.989536i \(0.546088\pi\)
\(380\) 0 0
\(381\) 21526.3 2.89456
\(382\) 0 0
\(383\) −1432.60 + 2481.34i −0.191129 + 0.331046i −0.945625 0.325260i \(-0.894548\pi\)
0.754495 + 0.656305i \(0.227882\pi\)
\(384\) 0 0
\(385\) 30.0272 52.0086i 0.00397488 0.00688469i
\(386\) 0 0
\(387\) −8008.58 −1.05194
\(388\) 0 0
\(389\) 6416.83 + 11114.3i 0.836365 + 1.44863i 0.892914 + 0.450227i \(0.148657\pi\)
−0.0565491 + 0.998400i \(0.518010\pi\)
\(390\) 0 0
\(391\) −297.473 −0.0384754
\(392\) 0 0
\(393\) 5103.01 + 8838.68i 0.654995 + 1.13448i
\(394\) 0 0
\(395\) 72.4526 + 125.492i 0.00922908 + 0.0159852i
\(396\) 0 0
\(397\) −1933.68 + 3349.23i −0.244454 + 0.423408i −0.961978 0.273127i \(-0.911942\pi\)
0.717524 + 0.696534i \(0.245276\pi\)
\(398\) 0 0
\(399\) 2963.95 + 5372.88i 0.371887 + 0.674137i
\(400\) 0 0
\(401\) −934.292 + 1618.24i −0.116350 + 0.201524i −0.918319 0.395842i \(-0.870453\pi\)
0.801969 + 0.597366i \(0.203786\pi\)
\(402\) 0 0
\(403\) −251.506 435.621i −0.0310878 0.0538457i
\(404\) 0 0
\(405\) −30.2408 52.3786i −0.00371031 0.00642645i
\(406\) 0 0
\(407\) 7048.92 0.858482
\(408\) 0 0
\(409\) −3666.90 6351.26i −0.443317 0.767848i 0.554616 0.832106i \(-0.312865\pi\)
−0.997933 + 0.0642584i \(0.979532\pi\)
\(410\) 0 0
\(411\) −16416.8 −1.97027
\(412\) 0 0
\(413\) 3969.67 6875.67i 0.472965 0.819200i
\(414\) 0 0
\(415\) −21.2416 + 36.7916i −0.00251256 + 0.00435188i
\(416\) 0 0
\(417\) 3688.89 0.433204
\(418\) 0 0
\(419\) 1286.23 0.149968 0.0749840 0.997185i \(-0.476109\pi\)
0.0749840 + 0.997185i \(0.476109\pi\)
\(420\) 0 0
\(421\) −757.659 + 1312.30i −0.0877103 + 0.151919i −0.906543 0.422114i \(-0.861288\pi\)
0.818833 + 0.574032i \(0.194622\pi\)
\(422\) 0 0
\(423\) 6173.78 10693.3i 0.709644 1.22914i
\(424\) 0 0
\(425\) −240.839 −0.0274880
\(426\) 0 0
\(427\) −705.324 1221.66i −0.0799368 0.138455i
\(428\) 0 0
\(429\) −16661.2 −1.87508
\(430\) 0 0
\(431\) −3663.73 6345.76i −0.409456 0.709199i 0.585373 0.810764i \(-0.300948\pi\)
−0.994829 + 0.101566i \(0.967615\pi\)
\(432\) 0 0
\(433\) 3511.74 + 6082.50i 0.389753 + 0.675073i 0.992416 0.122923i \(-0.0392269\pi\)
−0.602663 + 0.797996i \(0.705894\pi\)
\(434\) 0 0
\(435\) 28.0255 48.5416i 0.00308901 0.00535033i
\(436\) 0 0
\(437\) −12782.5 249.166i −1.39925 0.0272751i
\(438\) 0 0
\(439\) −7556.49 + 13088.2i −0.821530 + 1.42293i 0.0830132 + 0.996548i \(0.473546\pi\)
−0.904543 + 0.426383i \(0.859788\pi\)
\(440\) 0 0
\(441\) 4446.28 + 7701.19i 0.480108 + 0.831572i
\(442\) 0 0
\(443\) 983.331 + 1703.18i 0.105462 + 0.182665i 0.913927 0.405879i \(-0.133035\pi\)
−0.808465 + 0.588544i \(0.799701\pi\)
\(444\) 0 0
\(445\) 136.471 0.0145378
\(446\) 0 0
\(447\) −9536.43 16517.6i −1.00908 1.74777i
\(448\) 0 0
\(449\) −8276.44 −0.869909 −0.434954 0.900452i \(-0.643236\pi\)
−0.434954 + 0.900452i \(0.643236\pi\)
\(450\) 0 0
\(451\) 6251.14 10827.3i 0.652671 1.13046i
\(452\) 0 0
\(453\) 13714.5 23754.2i 1.42244 2.46373i
\(454\) 0 0
\(455\) 56.1894 0.00578945
\(456\) 0 0
\(457\) −16789.0 −1.71850 −0.859251 0.511555i \(-0.829070\pi\)
−0.859251 + 0.511555i \(0.829070\pi\)
\(458\) 0 0
\(459\) −60.3346 + 104.503i −0.00613547 + 0.0106269i
\(460\) 0 0
\(461\) −5237.00 + 9070.75i −0.529092 + 0.916414i 0.470332 + 0.882489i \(0.344134\pi\)
−0.999424 + 0.0339251i \(0.989199\pi\)
\(462\) 0 0
\(463\) −3416.87 −0.342970 −0.171485 0.985187i \(-0.554857\pi\)
−0.171485 + 0.985187i \(0.554857\pi\)
\(464\) 0 0
\(465\) 5.96678 + 10.3348i 0.000595060 + 0.00103067i
\(466\) 0 0
\(467\) 5081.31 0.503501 0.251751 0.967792i \(-0.418994\pi\)
0.251751 + 0.967792i \(0.418994\pi\)
\(468\) 0 0
\(469\) 4321.34 + 7484.78i 0.425461 + 0.736919i
\(470\) 0 0
\(471\) −13112.3 22711.2i −1.28277 2.22182i
\(472\) 0 0
\(473\) −5448.61 + 9437.27i −0.529656 + 0.917391i
\(474\) 0 0
\(475\) −10348.9 201.728i −0.999666 0.0194861i
\(476\) 0 0
\(477\) 3856.00 6678.79i 0.370135 0.641092i
\(478\) 0 0
\(479\) −5528.06 9574.89i −0.527315 0.913336i −0.999493 0.0318328i \(-0.989866\pi\)
0.472179 0.881503i \(-0.343468\pi\)
\(480\) 0 0
\(481\) 3297.63 + 5711.67i 0.312597 + 0.541434i
\(482\) 0 0
\(483\) 11437.7 1.07750
\(484\) 0 0
\(485\) −72.0600 124.812i −0.00674655 0.0116854i
\(486\) 0 0
\(487\) 18123.1 1.68632 0.843158 0.537667i \(-0.180694\pi\)
0.843158 + 0.537667i \(0.180694\pi\)
\(488\) 0 0
\(489\) −10272.9 + 17793.2i −0.950016 + 1.64548i
\(490\) 0 0
\(491\) −7620.44 + 13199.0i −0.700419 + 1.21316i 0.267901 + 0.963446i \(0.413670\pi\)
−0.968320 + 0.249714i \(0.919663\pi\)
\(492\) 0 0
\(493\) 102.302 0.00934571
\(494\) 0 0
\(495\) 223.015 0.0202501
\(496\) 0 0
\(497\) −4640.84 + 8038.17i −0.418853 + 0.725475i
\(498\) 0 0
\(499\) 4334.55 7507.67i 0.388860 0.673526i −0.603436 0.797411i \(-0.706202\pi\)
0.992297 + 0.123885i \(0.0395355\pi\)
\(500\) 0 0
\(501\) −1532.47 −0.136659
\(502\) 0 0
\(503\) 8232.19 + 14258.6i 0.729732 + 1.26393i 0.956997 + 0.290099i \(0.0936883\pi\)
−0.227265 + 0.973833i \(0.572978\pi\)
\(504\) 0 0
\(505\) 172.237 0.0151772
\(506\) 0 0
\(507\) 852.058 + 1475.81i 0.0746376 + 0.129276i
\(508\) 0 0
\(509\) 4912.20 + 8508.18i 0.427759 + 0.740900i 0.996674 0.0814962i \(-0.0259699\pi\)
−0.568915 + 0.822397i \(0.692637\pi\)
\(510\) 0 0
\(511\) −4343.31 + 7522.83i −0.376001 + 0.651253i
\(512\) 0 0
\(513\) −2680.13 + 4439.98i −0.230664 + 0.382124i
\(514\) 0 0
\(515\) 32.9711 57.1076i 0.00282113 0.00488634i
\(516\) 0 0
\(517\) −8400.61 14550.3i −0.714620 1.23776i
\(518\) 0 0
\(519\) 2301.35 + 3986.05i 0.194639 + 0.337125i
\(520\) 0 0
\(521\) 10650.5 0.895595 0.447798 0.894135i \(-0.352208\pi\)
0.447798 + 0.894135i \(0.352208\pi\)
\(522\) 0 0
\(523\) −5825.64 10090.3i −0.487070 0.843629i 0.512820 0.858496i \(-0.328601\pi\)
−0.999889 + 0.0148670i \(0.995268\pi\)
\(524\) 0 0
\(525\) 9260.10 0.769798
\(526\) 0 0
\(527\) −10.8903 + 18.8625i −0.000900168 + 0.00155914i
\(528\) 0 0
\(529\) −5831.88 + 10101.1i −0.479320 + 0.830206i
\(530\) 0 0
\(531\) 29483.2 2.40953
\(532\) 0 0
\(533\) 11697.7 0.950622
\(534\) 0 0
\(535\) 20.2798 35.1257i 0.00163883 0.00283853i
\(536\) 0 0
\(537\) 5767.22 9989.12i 0.463452 0.802723i
\(538\) 0 0
\(539\) 12100.0 0.966950
\(540\) 0 0
\(541\) −4978.81 8623.56i −0.395667 0.685316i 0.597519 0.801855i \(-0.296153\pi\)
−0.993186 + 0.116539i \(0.962820\pi\)
\(542\) 0 0
\(543\) −18561.9 −1.46697
\(544\) 0 0
\(545\) −52.6820 91.2480i −0.00414064 0.00717180i
\(546\) 0 0
\(547\) −193.113 334.481i −0.0150949 0.0261451i 0.858379 0.513016i \(-0.171472\pi\)
−0.873474 + 0.486870i \(0.838138\pi\)
\(548\) 0 0
\(549\) 2619.26 4536.69i 0.203620 0.352680i
\(550\) 0 0
\(551\) 4395.94 + 85.6885i 0.339879 + 0.00662515i
\(552\) 0 0
\(553\) 5084.43 8806.50i 0.390980 0.677198i
\(554\) 0 0
\(555\) −78.2338 135.505i −0.00598350 0.0103637i
\(556\) 0 0
\(557\) 4092.54 + 7088.49i 0.311322 + 0.539226i 0.978649 0.205539i \(-0.0658948\pi\)
−0.667327 + 0.744765i \(0.732561\pi\)
\(558\) 0 0
\(559\) −10195.9 −0.771450
\(560\) 0 0
\(561\) 360.717 + 624.781i 0.0271471 + 0.0470201i
\(562\) 0 0
\(563\) 9871.86 0.738986 0.369493 0.929234i \(-0.379531\pi\)
0.369493 + 0.929234i \(0.379531\pi\)
\(564\) 0 0
\(565\) −11.5227 + 19.9579i −0.000857988 + 0.00148608i
\(566\) 0 0
\(567\) −2122.18 + 3675.72i −0.157184 + 0.272250i
\(568\) 0 0
\(569\) −12867.7 −0.948054 −0.474027 0.880510i \(-0.657200\pi\)
−0.474027 + 0.880510i \(0.657200\pi\)
\(570\) 0 0
\(571\) 2729.67 0.200058 0.100029 0.994985i \(-0.468106\pi\)
0.100029 + 0.994985i \(0.468106\pi\)
\(572\) 0 0
\(573\) 17092.4 29604.8i 1.24615 2.15839i
\(574\) 0 0
\(575\) −9646.87 + 16708.9i −0.699656 + 1.21184i
\(576\) 0 0
\(577\) 8624.00 0.622222 0.311111 0.950374i \(-0.399299\pi\)
0.311111 + 0.950374i \(0.399299\pi\)
\(578\) 0 0
\(579\) 3159.35 + 5472.15i 0.226767 + 0.392772i
\(580\) 0 0
\(581\) 2981.31 0.212884
\(582\) 0 0
\(583\) −5246.83 9087.78i −0.372730 0.645587i
\(584\) 0 0
\(585\) 104.331 + 180.707i 0.00737361 + 0.0127715i
\(586\) 0 0
\(587\) 1995.20 3455.80i 0.140291 0.242991i −0.787315 0.616551i \(-0.788529\pi\)
0.927606 + 0.373559i \(0.121863\pi\)
\(588\) 0 0
\(589\) −483.759 + 801.408i −0.0338420 + 0.0560636i
\(590\) 0 0
\(591\) 632.958 1096.32i 0.0440548 0.0763052i
\(592\) 0 0
\(593\) −9617.19 16657.5i −0.665987 1.15352i −0.979017 0.203780i \(-0.934677\pi\)
0.313029 0.949743i \(-0.398656\pi\)
\(594\) 0 0
\(595\) −1.21651 2.10706i −8.38186e−5 0.000145178i
\(596\) 0 0
\(597\) −11169.9 −0.765749
\(598\) 0 0
\(599\) 11156.9 + 19324.3i 0.761033 + 1.31815i 0.942319 + 0.334717i \(0.108641\pi\)
−0.181286 + 0.983430i \(0.558026\pi\)
\(600\) 0 0
\(601\) 10725.0 0.727923 0.363962 0.931414i \(-0.381424\pi\)
0.363962 + 0.931414i \(0.381424\pi\)
\(602\) 0 0
\(603\) −16047.5 + 27795.2i −1.08376 + 1.87712i
\(604\) 0 0
\(605\) 62.4613 108.186i 0.00419738 0.00727007i
\(606\) 0 0
\(607\) 25360.0 1.69577 0.847884 0.530181i \(-0.177876\pi\)
0.847884 + 0.530181i \(0.177876\pi\)
\(608\) 0 0
\(609\) −3933.44 −0.261726
\(610\) 0 0
\(611\) 7859.97 13613.9i 0.520426 0.901404i
\(612\) 0 0
\(613\) 3946.86 6836.16i 0.260052 0.450424i −0.706203 0.708009i \(-0.749593\pi\)
0.966256 + 0.257585i \(0.0829268\pi\)
\(614\) 0 0
\(615\) −277.518 −0.0181961
\(616\) 0 0
\(617\) 6594.61 + 11422.2i 0.430290 + 0.745284i 0.996898 0.0787034i \(-0.0250780\pi\)
−0.566608 + 0.823987i \(0.691745\pi\)
\(618\) 0 0
\(619\) −1559.76 −0.101280 −0.0506399 0.998717i \(-0.516126\pi\)
−0.0506399 + 0.998717i \(0.516126\pi\)
\(620\) 0 0
\(621\) 4833.44 + 8371.77i 0.312334 + 0.540978i
\(622\) 0 0
\(623\) −4788.49 8293.91i −0.307940 0.533368i
\(624\) 0 0
\(625\) −7809.13 + 13525.8i −0.499784 + 0.865651i
\(626\) 0 0
\(627\) 14976.8 + 27149.2i 0.953935 + 1.72924i
\(628\) 0 0
\(629\) 142.789 247.317i 0.00905144 0.0156776i
\(630\) 0 0
\(631\) −3801.01 6583.54i −0.239803 0.415351i 0.720855 0.693086i \(-0.243750\pi\)
−0.960658 + 0.277735i \(0.910416\pi\)
\(632\) 0 0
\(633\) −11427.7 19793.3i −0.717550 1.24283i
\(634\) 0 0
\(635\) −366.833 −0.0229249
\(636\) 0 0
\(637\) 5660.66 + 9804.55i 0.352093 + 0.609843i
\(638\) 0 0
\(639\) −34468.0 −2.13386
\(640\) 0 0
\(641\) −9450.80 + 16369.3i −0.582346 + 1.00865i 0.412854 + 0.910797i \(0.364532\pi\)
−0.995201 + 0.0978565i \(0.968801\pi\)
\(642\) 0 0
\(643\) −14237.5 + 24660.1i −0.873210 + 1.51244i −0.0145519 + 0.999894i \(0.504632\pi\)
−0.858658 + 0.512549i \(0.828701\pi\)
\(644\) 0 0
\(645\) 241.890 0.0147665
\(646\) 0 0
\(647\) −18625.7 −1.13177 −0.565883 0.824485i \(-0.691465\pi\)
−0.565883 + 0.824485i \(0.691465\pi\)
\(648\) 0 0
\(649\) 20058.8 34742.8i 1.21321 2.10135i
\(650\) 0 0
\(651\) 418.725 725.253i 0.0252091 0.0436635i
\(652\) 0 0
\(653\) 3276.84 0.196374 0.0981872 0.995168i \(-0.468696\pi\)
0.0981872 + 0.995168i \(0.468696\pi\)
\(654\) 0 0
\(655\) −86.9611 150.621i −0.00518756 0.00898512i
\(656\) 0 0
\(657\) −32258.2 −1.91554
\(658\) 0 0
\(659\) 3275.97 + 5674.15i 0.193648 + 0.335408i 0.946456 0.322832i \(-0.104635\pi\)
−0.752809 + 0.658239i \(0.771301\pi\)
\(660\) 0 0
\(661\) −13591.2 23540.7i −0.799754 1.38522i −0.919776 0.392444i \(-0.871630\pi\)
0.120021 0.992771i \(-0.461704\pi\)
\(662\) 0 0
\(663\) −337.502 + 584.571i −0.0197700 + 0.0342426i
\(664\) 0 0
\(665\) −50.5090 91.5600i −0.00294535 0.00533916i
\(666\) 0 0
\(667\) 4097.73 7097.47i 0.237878 0.412017i
\(668\) 0 0
\(669\) −14060.6 24353.8i −0.812580 1.40743i
\(670\) 0 0
\(671\) −3564.01 6173.04i −0.205048 0.355153i
\(672\) 0 0
\(673\) −14256.8 −0.816582 −0.408291 0.912852i \(-0.633875\pi\)
−0.408291 + 0.912852i \(0.633875\pi\)
\(674\) 0 0
\(675\) 3913.23 + 6777.91i 0.223141 + 0.386492i
\(676\) 0 0
\(677\) 27947.0 1.58655 0.793273 0.608866i \(-0.208375\pi\)
0.793273 + 0.608866i \(0.208375\pi\)
\(678\) 0 0
\(679\) −5056.88 + 8758.78i −0.285811 + 0.495039i
\(680\) 0 0
\(681\) −760.967 + 1318.03i −0.0428199 + 0.0741662i
\(682\) 0 0
\(683\) 10455.7 0.585761 0.292880 0.956149i \(-0.405386\pi\)
0.292880 + 0.956149i \(0.405386\pi\)
\(684\) 0 0
\(685\) 279.760 0.0156045
\(686\) 0 0
\(687\) −3266.97 + 5658.56i −0.181431 + 0.314247i
\(688\) 0 0
\(689\) 4909.16 8502.91i 0.271443 0.470152i
\(690\) 0 0
\(691\) −6684.98 −0.368030 −0.184015 0.982923i \(-0.558909\pi\)
−0.184015 + 0.982923i \(0.558909\pi\)
\(692\) 0 0
\(693\) −7825.16 13553.6i −0.428937 0.742940i
\(694\) 0 0
\(695\) −62.8630 −0.00343098
\(696\) 0 0
\(697\) −253.256 438.653i −0.0137629 0.0238381i
\(698\) 0 0
\(699\) −20002.0 34644.5i −1.08233 1.87464i
\(700\) 0 0
\(701\) −1109.57 + 1921.83i −0.0597829 + 0.103547i −0.894368 0.447332i \(-0.852374\pi\)
0.834585 + 0.550879i \(0.185708\pi\)
\(702\) 0 0
\(703\) 6342.84 10507.7i 0.340291 0.563735i
\(704\) 0 0
\(705\) −186.472 + 322.978i −0.00996160 + 0.0172540i
\(706\) 0 0
\(707\) −6043.47 10467.6i −0.321482 0.556824i
\(708\) 0 0
\(709\) −10880.2 18845.0i −0.576323 0.998220i −0.995897 0.0904991i \(-0.971154\pi\)
0.419574 0.907721i \(-0.362180\pi\)
\(710\) 0 0
\(711\) 37762.7 1.99186
\(712\) 0 0
\(713\) 872.428 + 1511.09i 0.0458242 + 0.0793699i
\(714\) 0 0
\(715\) 283.925 0.0148506
\(716\) 0 0
\(717\) −26242.7 + 45453.7i −1.36688 + 2.36750i
\(718\) 0 0
\(719\) −8761.18 + 15174.8i −0.454433 + 0.787100i −0.998655 0.0518405i \(-0.983491\pi\)
0.544223 + 0.838941i \(0.316825\pi\)
\(720\) 0 0
\(721\) −4627.56 −0.239028
\(722\) 0 0
\(723\) −1416.66 −0.0728714
\(724\) 0 0
\(725\) 3317.58 5746.22i 0.169947 0.294357i
\(726\) 0 0
\(727\) −8678.76 + 15032.1i −0.442748 + 0.766861i −0.997892 0.0648925i \(-0.979330\pi\)
0.555145 + 0.831754i \(0.312663\pi\)
\(728\) 0 0
\(729\) −29069.9 −1.47691
\(730\) 0 0
\(731\) 220.743 + 382.338i 0.0111689 + 0.0193451i
\(732\) 0 0
\(733\) 31302.6 1.57734 0.788669 0.614818i \(-0.210770\pi\)
0.788669 + 0.614818i \(0.210770\pi\)
\(734\) 0 0
\(735\) −134.295 232.605i −0.00673951 0.0116732i
\(736\) 0 0
\(737\) 21835.8 + 37820.7i 1.09136 + 1.89029i
\(738\) 0 0
\(739\) 8352.91 14467.7i 0.415787 0.720165i −0.579724 0.814813i \(-0.696839\pi\)
0.995511 + 0.0946487i \(0.0301728\pi\)
\(740\) 0 0
\(741\) −14992.2 + 24836.5i −0.743257 + 1.23130i
\(742\) 0 0
\(743\) 5866.80 10161.6i 0.289679 0.501740i −0.684054 0.729432i \(-0.739785\pi\)
0.973733 + 0.227692i \(0.0731180\pi\)
\(744\) 0 0
\(745\) 162.512 + 281.478i 0.00799190 + 0.0138424i
\(746\) 0 0
\(747\) 5535.62 + 9587.98i 0.271135 + 0.469620i
\(748\) 0 0
\(749\) −2846.31 −0.138855
\(750\) 0 0
\(751\) −5718.63 9904.95i −0.277864 0.481274i 0.692990 0.720947i \(-0.256293\pi\)
−0.970854 + 0.239673i \(0.922960\pi\)
\(752\) 0 0
\(753\) 39565.8 1.91482
\(754\) 0 0
\(755\) −233.711 + 404.799i −0.0112657 + 0.0195128i
\(756\) 0 0
\(757\) 13329.0 23086.5i 0.639960 1.10844i −0.345481 0.938426i \(-0.612284\pi\)
0.985441 0.170018i \(-0.0543826\pi\)
\(758\) 0 0
\(759\) 57794.6 2.76391
\(760\) 0 0
\(761\) −29934.0 −1.42590 −0.712948 0.701217i \(-0.752640\pi\)
−0.712948 + 0.701217i \(0.752640\pi\)
\(762\) 0 0
\(763\) −3697.02 + 6403.42i −0.175414 + 0.303826i
\(764\) 0 0
\(765\) 4.51758 7.82468i 0.000213508 0.000369806i
\(766\) 0 0
\(767\) 37535.6 1.76706
\(768\) 0 0
\(769\) 15271.1 + 26450.3i 0.716112 + 1.24034i 0.962529 + 0.271178i \(0.0874132\pi\)
−0.246418 + 0.969164i \(0.579253\pi\)
\(770\) 0 0
\(771\) 29459.6 1.37609
\(772\) 0 0
\(773\) 5386.26 + 9329.28i 0.250621 + 0.434089i 0.963697 0.266998i \(-0.0860317\pi\)
−0.713076 + 0.701087i \(0.752698\pi\)
\(774\) 0 0
\(775\) 706.331 + 1223.40i 0.0327383 + 0.0567043i
\(776\) 0 0
\(777\) −5490.14 + 9509.20i −0.253485 + 0.439048i
\(778\) 0 0
\(779\) −10515.1 19061.2i −0.483623 0.876686i
\(780\) 0 0
\(781\) −23450.2 + 40616.9i −1.07441 + 1.86093i
\(782\) 0 0
\(783\) −1662.23 2879.07i −0.0758663 0.131404i
\(784\) 0 0
\(785\) 223.449 + 387.025i 0.0101595 + 0.0175968i
\(786\) 0 0
\(787\) −9202.56 −0.416818 −0.208409 0.978042i \(-0.566829\pi\)
−0.208409 + 0.978042i \(0.566829\pi\)
\(788\) 0 0
\(789\) 989.479 + 1713.83i 0.0446469 + 0.0773306i
\(790\) 0 0
\(791\) 1617.23 0.0726956
\(792\) 0 0
\(793\) 3334.63 5775.76i 0.149327 0.258642i
\(794\) 0 0
\(795\) −116.466 + 201.725i −0.00519575 + 0.00899931i
\(796\) 0 0
\(797\) 23453.4 1.04236 0.521181 0.853446i \(-0.325492\pi\)
0.521181 + 0.853446i \(0.325492\pi\)
\(798\) 0 0
\(799\) −680.679 −0.0301385
\(800\) 0 0
\(801\) 17782.3 30799.9i 0.784404 1.35863i
\(802\) 0 0
\(803\) −21946.7 + 38012.9i −0.964488 + 1.67054i
\(804\) 0 0
\(805\) −194.911 −0.00853379
\(806\) 0 0
\(807\) −8542.94 14796.8i −0.372646 0.645443i
\(808\) 0 0
\(809\) −22322.4 −0.970105 −0.485052 0.874485i \(-0.661199\pi\)
−0.485052 + 0.874485i \(0.661199\pi\)
\(810\) 0 0
\(811\) −8031.11 13910.3i −0.347732 0.602289i 0.638114 0.769942i \(-0.279715\pi\)
−0.985846 + 0.167652i \(0.946381\pi\)
\(812\) 0 0
\(813\) −5274.44 9135.60i −0.227531 0.394095i
\(814\) 0 0
\(815\) 175.062 303.217i 0.00752413 0.0130322i
\(816\) 0 0
\(817\) 9165.16 + 16614.1i 0.392470 + 0.711449i
\(818\) 0 0
\(819\) 7321.55 12681.3i 0.312376 0.541050i
\(820\) 0 0
\(821\) −15725.2 27236.8i −0.668467 1.15782i −0.978333 0.207039i \(-0.933617\pi\)
0.309865 0.950780i \(-0.399716\pi\)
\(822\) 0 0
\(823\) −4638.10 8033.42i −0.196445 0.340252i 0.750929 0.660383i \(-0.229606\pi\)
−0.947373 + 0.320132i \(0.896273\pi\)
\(824\) 0 0
\(825\) 46791.4 1.97463
\(826\) 0 0
\(827\) −16678.2 28887.5i −0.701281 1.21465i −0.968017 0.250884i \(-0.919279\pi\)
0.266737 0.963769i \(-0.414055\pi\)
\(828\) 0 0
\(829\) 28257.5 1.18386 0.591932 0.805988i \(-0.298365\pi\)
0.591932 + 0.805988i \(0.298365\pi\)
\(830\) 0 0
\(831\) −754.726 + 1307.22i −0.0315056 + 0.0545693i
\(832\) 0 0
\(833\) 245.109 424.540i 0.0101951 0.0176584i
\(834\) 0 0
\(835\) 26.1151 0.00108234
\(836\) 0 0
\(837\) 707.796 0.0292294
\(838\) 0 0
\(839\) −4415.75 + 7648.30i −0.181703 + 0.314718i −0.942460 0.334318i \(-0.891494\pi\)
0.760758 + 0.649036i \(0.224827\pi\)
\(840\) 0 0
\(841\) 10785.3 18680.7i 0.442219 0.765946i
\(842\) 0 0
\(843\) 2235.43 0.0913314
\(844\) 0 0
\(845\) −14.5200 25.1495i −0.000591130 0.00102387i
\(846\) 0 0
\(847\) −8766.58 −0.355635
\(848\) 0 0
\(849\) −353.564 612.392i −0.0142925 0.0247553i
\(850\) 0 0
\(851\) −11438.9 19812.7i −0.460776 0.798087i
\(852\) 0 0
\(853\) −16810.4 + 29116.6i −0.674770 + 1.16874i 0.301766 + 0.953382i \(0.402424\pi\)
−0.976536 + 0.215354i \(0.930909\pi\)
\(854\) 0 0
\(855\) 200.676 332.445i 0.00802687 0.0132975i
\(856\) 0 0
\(857\) 11754.4 20359.2i 0.468520 0.811500i −0.530833 0.847476i \(-0.678121\pi\)
0.999353 + 0.0359766i \(0.0114542\pi\)
\(858\) 0 0
\(859\) −5258.38 9107.79i −0.208863 0.361762i 0.742493 0.669853i \(-0.233643\pi\)
−0.951357 + 0.308091i \(0.900310\pi\)
\(860\) 0 0
\(861\) 9737.55 + 16865.9i 0.385430 + 0.667584i
\(862\) 0 0
\(863\) −9794.63 −0.386342 −0.193171 0.981165i \(-0.561877\pi\)
−0.193171 + 0.981165i \(0.561877\pi\)
\(864\) 0 0
\(865\) −39.2175 67.9268i −0.00154154 0.00267003i
\(866\) 0 0
\(867\) −38641.9 −1.51367
\(868\) 0 0
\(869\) 25691.7 44499.3i 1.00291 1.73709i
\(870\) 0 0
\(871\) −20430.5 + 35386.6i −0.794787 + 1.37661i
\(872\) 0 0
\(873\) −37558.0 −1.45607
\(874\) 0 0
\(875\) −315.628 −0.0121945
\(876\) 0 0
\(877\) −24459.4 + 42364.9i −0.941773 + 1.63120i −0.179686 + 0.983724i \(0.557508\pi\)
−0.762087 + 0.647474i \(0.775825\pi\)
\(878\) 0 0
\(879\) 23447.1 40611.5i 0.899716 1.55835i
\(880\) 0 0
\(881\) −37028.4 −1.41603 −0.708013 0.706199i \(-0.750408\pi\)
−0.708013 + 0.706199i \(0.750408\pi\)
\(882\) 0 0
\(883\) 16940.7 + 29342.1i 0.645639 + 1.11828i 0.984154 + 0.177318i \(0.0567421\pi\)
−0.338515 + 0.940961i \(0.609925\pi\)
\(884\) 0 0
\(885\) −890.505 −0.0338237
\(886\) 0 0
\(887\) 15256.1 + 26424.4i 0.577509 + 1.00027i 0.995764 + 0.0919448i \(0.0293083\pi\)
−0.418256 + 0.908329i \(0.637358\pi\)
\(888\) 0 0
\(889\) 12871.5 + 22294.0i 0.485596 + 0.841077i
\(890\) 0 0
\(891\) −10723.4 + 18573.4i −0.403195 + 0.698354i
\(892\) 0 0
\(893\) −29249.0 570.141i −1.09606 0.0213651i
\(894\) 0 0
\(895\) −98.2800 + 170.226i −0.00367055 + 0.00635757i
\(896\) 0 0
\(897\) 27037.5 + 46830.4i 1.00642 + 1.74317i
\(898\) 0 0
\(899\) −300.030 519.667i −0.0111308 0.0192790i
\(900\) 0 0
\(901\) −425.137 −0.0157196
\(902\) 0 0
\(903\) −8487.43 14700.7i −0.312784 0.541758i
\(904\) 0 0
\(905\) 316.315 0.0116184
\(906\) 0 0
\(907\) −4101.70 + 7104.35i −0.150160 + 0.260084i −0.931286 0.364289i \(-0.881312\pi\)
0.781126 + 0.624373i \(0.214645\pi\)
\(908\) 0 0
\(909\) 22442.8 38872.0i 0.818899 1.41837i
\(910\) 0 0
\(911\) −26297.5 −0.956395 −0.478198 0.878252i \(-0.658710\pi\)
−0.478198 + 0.878252i \(0.658710\pi\)
\(912\) 0 0
\(913\) 15064.6 0.546073
\(914\) 0 0
\(915\) −79.1117 + 137.025i −0.00285831 + 0.00495073i
\(916\) 0 0
\(917\) −6102.59 + 10570.0i −0.219766 + 0.380645i
\(918\) 0 0
\(919\) 39244.1 1.40864 0.704321 0.709882i \(-0.251252\pi\)
0.704321 + 0.709882i \(0.251252\pi\)
\(920\) 0 0
\(921\) −8873.63 15369.6i −0.317477 0.549886i
\(922\) 0 0
\(923\) −43881.9 −1.56489
\(924\) 0 0
\(925\) −9261.10 16040.7i −0.329192 0.570178i
\(926\) 0 0
\(927\) −8592.36 14882.4i −0.304434 0.527295i
\(928\) 0 0
\(929\) 22995.1 39828.6i 0.812103 1.40660i −0.0992873 0.995059i \(-0.531656\pi\)
0.911390 0.411544i \(-0.135010\pi\)
\(930\) 0 0
\(931\) 10888.0 18037.3i 0.383287 0.634963i
\(932\) 0 0
\(933\) −23334.3 + 40416.1i −0.818788 + 1.41818i
\(934\) 0 0
\(935\) −6.14704 10.6470i −0.000215005 0.000372399i
\(936\) 0 0
\(937\) 286.074 + 495.494i 0.00997399 + 0.0172755i 0.870969 0.491337i \(-0.163492\pi\)
−0.860995 + 0.508613i \(0.830158\pi\)
\(938\) 0 0
\(939\) −47550.7 −1.65257
\(940\) 0 0
\(941\) 27767.8 + 48095.2i 0.961960 + 1.66616i 0.717570 + 0.696486i \(0.245254\pi\)
0.244389 + 0.969677i \(0.421413\pi\)
\(942\) 0 0
\(943\) −40577.1 −1.40124
\(944\) 0 0
\(945\) −39.5325 + 68.4723i −0.00136084 + 0.00235704i
\(946\) 0 0
\(947\) −1469.65 + 2545.51i −0.0504300 + 0.0873473i −0.890138 0.455690i \(-0.849393\pi\)
0.839708 + 0.543037i \(0.182726\pi\)
\(948\) 0 0
\(949\) −41068.6 −1.40479
\(950\) 0 0
\(951\) 48551.7 1.65552
\(952\) 0 0
\(953\) 11592.8 20079.4i 0.394049 0.682513i −0.598930 0.800801i \(-0.704407\pi\)
0.992979 + 0.118288i \(0.0377407\pi\)
\(954\) 0 0
\(955\) −291.273 + 504.500i −0.00986951 + 0.0170945i
\(956\) 0 0
\(957\) −19875.7 −0.671358
\(958\) 0 0
\(959\) −9816.24 17002.2i −0.330535 0.572503i
\(960\) 0 0
\(961\) −29663.2 −0.995712
\(962\) 0 0
\(963\) −5284.97 9153.84i −0.176849 0.306312i
\(964\) 0 0
\(965\) −53.8389 93.2516i −0.00179599 0.00311075i
\(966\) 0 0
\(967\) 8276.81 14335.9i 0.275247 0.476742i −0.694950 0.719058i \(-0.744574\pi\)
0.970198 + 0.242315i \(0.0779069\pi\)
\(968\) 0 0
\(969\) 1255.94 + 24.4815i 0.0416372 + 0.000811620i
\(970\) 0 0
\(971\) 10422.0 18051.4i 0.344445 0.596597i −0.640807 0.767702i \(-0.721400\pi\)
0.985253 + 0.171105i \(0.0547337\pi\)
\(972\) 0 0
\(973\) 2205.74 + 3820.45i 0.0726749 + 0.125877i
\(974\) 0 0
\(975\) 21890.0 + 37914.6i 0.719016 + 1.24537i
\(976\) 0 0
\(977\) 6747.90 0.220967 0.110483 0.993878i \(-0.464760\pi\)
0.110483 + 0.993878i \(0.464760\pi\)
\(978\) 0 0
\(979\) −24196.3 41909.2i −0.789904 1.36815i
\(980\) 0 0
\(981\) −27458.1 −0.893650
\(982\) 0 0
\(983\) 10992.1 19038.9i 0.356658 0.617750i −0.630742 0.775992i \(-0.717250\pi\)
0.987400 + 0.158243i \(0.0505829\pi\)
\(984\) 0 0
\(985\) −10.7863 + 18.6825i −0.000348915 + 0.000604338i
\(986\) 0 0
\(987\) 26171.7 0.844026
\(988\) 0 0
\(989\) 35367.7 1.13714
\(990\) 0 0
\(991\) 4924.35 8529.22i 0.157848 0.273400i −0.776245 0.630432i \(-0.782878\pi\)
0.934092 + 0.357032i \(0.116211\pi\)
\(992\) 0 0
\(993\) 27010.0 46782.7i 0.863179 1.49507i
\(994\) 0 0
\(995\) 190.347 0.00606473
\(996\) 0 0
\(997\) 20176.0 + 34945.8i 0.640902 + 1.11008i 0.985232 + 0.171226i \(0.0547729\pi\)
−0.344330 + 0.938849i \(0.611894\pi\)
\(998\) 0 0
\(999\) −9280.31 −0.293910
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 76.4.e.a.49.1 yes 10
3.2 odd 2 684.4.k.c.505.3 10
4.3 odd 2 304.4.i.f.49.5 10
19.7 even 3 inner 76.4.e.a.45.1 10
19.8 odd 6 1444.4.a.g.1.1 5
19.11 even 3 1444.4.a.f.1.5 5
57.26 odd 6 684.4.k.c.577.3 10
76.7 odd 6 304.4.i.f.273.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.e.a.45.1 10 19.7 even 3 inner
76.4.e.a.49.1 yes 10 1.1 even 1 trivial
304.4.i.f.49.5 10 4.3 odd 2
304.4.i.f.273.5 10 76.7 odd 6
684.4.k.c.505.3 10 3.2 odd 2
684.4.k.c.577.3 10 57.26 odd 6
1444.4.a.f.1.5 5 19.11 even 3
1444.4.a.g.1.1 5 19.8 odd 6