Properties

Label 567.4.c.c.566.12
Level $567$
Weight $4$
Character 567.566
Analytic conductor $33.454$
Analytic rank $0$
Dimension $44$
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] = [567,4,Mod(566,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.566");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 567.c (of order \(2\), degree \(1\), 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: \(33.4540829733\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 566.12
Character \(\chi\) \(=\) 567.566
Dual form 567.4.c.c.566.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.725795i q^{2} +7.47322 q^{4} +11.0664 q^{5} +(-17.8071 + 5.08977i) q^{7} +11.2304i q^{8} +O(q^{10})\) \(q+0.725795i q^{2} +7.47322 q^{4} +11.0664 q^{5} +(-17.8071 + 5.08977i) q^{7} +11.2304i q^{8} +8.03191i q^{10} +0.253624i q^{11} +14.3133i q^{13} +(-3.69413 - 12.9243i) q^{14} +51.6348 q^{16} -92.5854 q^{17} +130.104i q^{19} +82.7013 q^{20} -0.184079 q^{22} +118.700i q^{23} -2.53586 q^{25} -10.3885 q^{26} +(-133.077 + 38.0370i) q^{28} +286.736i q^{29} -247.365i q^{31} +127.319i q^{32} -67.1981i q^{34} +(-197.060 + 56.3252i) q^{35} +188.615 q^{37} -94.4288 q^{38} +124.279i q^{40} -106.487 q^{41} +43.1172 q^{43} +1.89539i q^{44} -86.1520 q^{46} +137.764 q^{47} +(291.188 - 181.269i) q^{49} -1.84052i q^{50} +106.966i q^{52} +419.280i q^{53} +2.80669i q^{55} +(-57.1602 - 199.981i) q^{56} -208.111 q^{58} +435.120 q^{59} -188.798i q^{61} +179.537 q^{62} +320.670 q^{64} +158.395i q^{65} +370.699 q^{67} -691.911 q^{68} +(-40.8806 - 143.025i) q^{70} +26.1630i q^{71} -728.545i q^{73} +136.896i q^{74} +972.295i q^{76} +(-1.29089 - 4.51632i) q^{77} +97.3557 q^{79} +571.409 q^{80} -77.2880i q^{82} -802.718 q^{83} -1024.58 q^{85} +31.2943i q^{86} -2.84830 q^{88} +236.618 q^{89} +(-72.8512 - 254.878i) q^{91} +887.072i q^{92} +99.9883i q^{94} +1439.77i q^{95} +1464.27i q^{97} +(131.564 + 211.343i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 44 q - 156 q^{4} - 10 q^{7} + 484 q^{16} + 68 q^{22} + 704 q^{25} + 300 q^{28} + 328 q^{37} + 340 q^{43} + 968 q^{46} + 158 q^{49} + 1076 q^{58} - 808 q^{64} + 1180 q^{67} - 768 q^{70} + 604 q^{79} + 1224 q^{85} - 2588 q^{88} + 210 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.725795i 0.256607i 0.991735 + 0.128304i \(0.0409532\pi\)
−0.991735 + 0.128304i \(0.959047\pi\)
\(3\) 0 0
\(4\) 7.47322 0.934153
\(5\) 11.0664 0.989805 0.494902 0.868949i \(-0.335204\pi\)
0.494902 + 0.868949i \(0.335204\pi\)
\(6\) 0 0
\(7\) −17.8071 + 5.08977i −0.961495 + 0.274822i
\(8\) 11.2304i 0.496318i
\(9\) 0 0
\(10\) 8.03191i 0.253991i
\(11\) 0.253624i 0.00695186i 0.999994 + 0.00347593i \(0.00110643\pi\)
−0.999994 + 0.00347593i \(0.998894\pi\)
\(12\) 0 0
\(13\) 14.3133i 0.305368i 0.988275 + 0.152684i \(0.0487916\pi\)
−0.988275 + 0.152684i \(0.951208\pi\)
\(14\) −3.69413 12.9243i −0.0705214 0.246727i
\(15\) 0 0
\(16\) 51.6348 0.806794
\(17\) −92.5854 −1.32090 −0.660449 0.750871i \(-0.729634\pi\)
−0.660449 + 0.750871i \(0.729634\pi\)
\(18\) 0 0
\(19\) 130.104i 1.57094i 0.618899 + 0.785470i \(0.287579\pi\)
−0.618899 + 0.785470i \(0.712421\pi\)
\(20\) 82.7013 0.924629
\(21\) 0 0
\(22\) −0.184079 −0.00178390
\(23\) 118.700i 1.07612i 0.842908 + 0.538058i \(0.180842\pi\)
−0.842908 + 0.538058i \(0.819158\pi\)
\(24\) 0 0
\(25\) −2.53586 −0.0202869
\(26\) −10.3885 −0.0783597
\(27\) 0 0
\(28\) −133.077 + 38.0370i −0.898183 + 0.256726i
\(29\) 286.736i 1.83605i 0.396522 + 0.918025i \(0.370217\pi\)
−0.396522 + 0.918025i \(0.629783\pi\)
\(30\) 0 0
\(31\) 247.365i 1.43316i −0.697503 0.716582i \(-0.745705\pi\)
0.697503 0.716582i \(-0.254295\pi\)
\(32\) 127.319i 0.703347i
\(33\) 0 0
\(34\) 67.1981i 0.338952i
\(35\) −197.060 + 56.3252i −0.951692 + 0.272020i
\(36\) 0 0
\(37\) 188.615 0.838058 0.419029 0.907973i \(-0.362371\pi\)
0.419029 + 0.907973i \(0.362371\pi\)
\(38\) −94.4288 −0.403115
\(39\) 0 0
\(40\) 124.279i 0.491258i
\(41\) −106.487 −0.405623 −0.202811 0.979218i \(-0.565008\pi\)
−0.202811 + 0.979218i \(0.565008\pi\)
\(42\) 0 0
\(43\) 43.1172 0.152914 0.0764572 0.997073i \(-0.475639\pi\)
0.0764572 + 0.997073i \(0.475639\pi\)
\(44\) 1.89539i 0.00649410i
\(45\) 0 0
\(46\) −86.1520 −0.276139
\(47\) 137.764 0.427551 0.213776 0.976883i \(-0.431424\pi\)
0.213776 + 0.976883i \(0.431424\pi\)
\(48\) 0 0
\(49\) 291.188 181.269i 0.848946 0.528480i
\(50\) 1.84052i 0.00520576i
\(51\) 0 0
\(52\) 106.966i 0.285260i
\(53\) 419.280i 1.08665i 0.839522 + 0.543325i \(0.182835\pi\)
−0.839522 + 0.543325i \(0.817165\pi\)
\(54\) 0 0
\(55\) 2.80669i 0.00688098i
\(56\) −57.1602 199.981i −0.136399 0.477207i
\(57\) 0 0
\(58\) −208.111 −0.471144
\(59\) 435.120 0.960132 0.480066 0.877232i \(-0.340613\pi\)
0.480066 + 0.877232i \(0.340613\pi\)
\(60\) 0 0
\(61\) 188.798i 0.396280i −0.980174 0.198140i \(-0.936510\pi\)
0.980174 0.198140i \(-0.0634901\pi\)
\(62\) 179.537 0.367761
\(63\) 0 0
\(64\) 320.670 0.626310
\(65\) 158.395i 0.302254i
\(66\) 0 0
\(67\) 370.699 0.675942 0.337971 0.941157i \(-0.390259\pi\)
0.337971 + 0.941157i \(0.390259\pi\)
\(68\) −691.911 −1.23392
\(69\) 0 0
\(70\) −40.8806 143.025i −0.0698024 0.244211i
\(71\) 26.1630i 0.0437321i 0.999761 + 0.0218661i \(0.00696074\pi\)
−0.999761 + 0.0218661i \(0.993039\pi\)
\(72\) 0 0
\(73\) 728.545i 1.16808i −0.811725 0.584039i \(-0.801471\pi\)
0.811725 0.584039i \(-0.198529\pi\)
\(74\) 136.896i 0.215052i
\(75\) 0 0
\(76\) 972.295i 1.46750i
\(77\) −1.29089 4.51632i −0.00191052 0.00668418i
\(78\) 0 0
\(79\) 97.3557 0.138650 0.0693251 0.997594i \(-0.477915\pi\)
0.0693251 + 0.997594i \(0.477915\pi\)
\(80\) 571.409 0.798568
\(81\) 0 0
\(82\) 77.2880i 0.104086i
\(83\) −802.718 −1.06156 −0.530781 0.847509i \(-0.678101\pi\)
−0.530781 + 0.847509i \(0.678101\pi\)
\(84\) 0 0
\(85\) −1024.58 −1.30743
\(86\) 31.2943i 0.0392390i
\(87\) 0 0
\(88\) −2.84830 −0.00345033
\(89\) 236.618 0.281814 0.140907 0.990023i \(-0.454998\pi\)
0.140907 + 0.990023i \(0.454998\pi\)
\(90\) 0 0
\(91\) −72.8512 254.878i −0.0839218 0.293610i
\(92\) 887.072i 1.00526i
\(93\) 0 0
\(94\) 99.9883i 0.109713i
\(95\) 1439.77i 1.55492i
\(96\) 0 0
\(97\) 1464.27i 1.53272i 0.642412 + 0.766360i \(0.277934\pi\)
−0.642412 + 0.766360i \(0.722066\pi\)
\(98\) 131.564 + 211.343i 0.135612 + 0.217846i
\(99\) 0 0
\(100\) −18.9510 −0.0189510
\(101\) −1567.44 −1.54422 −0.772112 0.635487i \(-0.780799\pi\)
−0.772112 + 0.635487i \(0.780799\pi\)
\(102\) 0 0
\(103\) 100.296i 0.0959461i 0.998849 + 0.0479730i \(0.0152762\pi\)
−0.998849 + 0.0479730i \(0.984724\pi\)
\(104\) −160.743 −0.151560
\(105\) 0 0
\(106\) −304.311 −0.278843
\(107\) 553.381i 0.499976i 0.968249 + 0.249988i \(0.0804266\pi\)
−0.968249 + 0.249988i \(0.919573\pi\)
\(108\) 0 0
\(109\) −400.600 −0.352023 −0.176012 0.984388i \(-0.556320\pi\)
−0.176012 + 0.984388i \(0.556320\pi\)
\(110\) −2.03708 −0.00176571
\(111\) 0 0
\(112\) −919.468 + 262.809i −0.775728 + 0.221725i
\(113\) 712.932i 0.593513i 0.954953 + 0.296756i \(0.0959050\pi\)
−0.954953 + 0.296756i \(0.904095\pi\)
\(114\) 0 0
\(115\) 1313.58i 1.06514i
\(116\) 2142.84i 1.71515i
\(117\) 0 0
\(118\) 315.808i 0.246377i
\(119\) 1648.68 471.239i 1.27004 0.363012i
\(120\) 0 0
\(121\) 1330.94 0.999952
\(122\) 137.029 0.101688
\(123\) 0 0
\(124\) 1848.61i 1.33879i
\(125\) −1411.36 −1.00988
\(126\) 0 0
\(127\) 102.170 0.0713865 0.0356932 0.999363i \(-0.488636\pi\)
0.0356932 + 0.999363i \(0.488636\pi\)
\(128\) 1251.30i 0.864063i
\(129\) 0 0
\(130\) −114.963 −0.0775608
\(131\) −472.845 −0.315364 −0.157682 0.987490i \(-0.550402\pi\)
−0.157682 + 0.987490i \(0.550402\pi\)
\(132\) 0 0
\(133\) −662.199 2316.78i −0.431729 1.51045i
\(134\) 269.052i 0.173452i
\(135\) 0 0
\(136\) 1039.77i 0.655585i
\(137\) 2619.41i 1.63351i 0.576983 + 0.816756i \(0.304230\pi\)
−0.576983 + 0.816756i \(0.695770\pi\)
\(138\) 0 0
\(139\) 1242.37i 0.758104i −0.925375 0.379052i \(-0.876250\pi\)
0.925375 0.379052i \(-0.123750\pi\)
\(140\) −1472.67 + 420.931i −0.889026 + 0.254108i
\(141\) 0 0
\(142\) −18.9890 −0.0112220
\(143\) −3.63018 −0.00212287
\(144\) 0 0
\(145\) 3173.12i 1.81733i
\(146\) 528.775 0.299738
\(147\) 0 0
\(148\) 1409.56 0.782874
\(149\) 1124.55i 0.618302i −0.951013 0.309151i \(-0.899955\pi\)
0.951013 0.309151i \(-0.100045\pi\)
\(150\) 0 0
\(151\) −475.597 −0.256315 −0.128157 0.991754i \(-0.540906\pi\)
−0.128157 + 0.991754i \(0.540906\pi\)
\(152\) −1461.12 −0.779686
\(153\) 0 0
\(154\) 3.27792 0.936921i 0.00171521 0.000490255i
\(155\) 2737.43i 1.41855i
\(156\) 0 0
\(157\) 1933.32i 0.982775i −0.870941 0.491388i \(-0.836490\pi\)
0.870941 0.491388i \(-0.163510\pi\)
\(158\) 70.6603i 0.0355787i
\(159\) 0 0
\(160\) 1408.96i 0.696176i
\(161\) −604.156 2113.71i −0.295740 1.03468i
\(162\) 0 0
\(163\) −743.323 −0.357187 −0.178594 0.983923i \(-0.557155\pi\)
−0.178594 + 0.983923i \(0.557155\pi\)
\(164\) −795.803 −0.378913
\(165\) 0 0
\(166\) 582.609i 0.272405i
\(167\) 3623.20 1.67887 0.839435 0.543460i \(-0.182886\pi\)
0.839435 + 0.543460i \(0.182886\pi\)
\(168\) 0 0
\(169\) 1992.13 0.906750
\(170\) 743.638i 0.335496i
\(171\) 0 0
\(172\) 322.225 0.142845
\(173\) 3297.00 1.44894 0.724470 0.689306i \(-0.242085\pi\)
0.724470 + 0.689306i \(0.242085\pi\)
\(174\) 0 0
\(175\) 45.1564 12.9069i 0.0195057 0.00557528i
\(176\) 13.0958i 0.00560872i
\(177\) 0 0
\(178\) 171.736i 0.0723155i
\(179\) 2746.77i 1.14695i −0.819225 0.573473i \(-0.805596\pi\)
0.819225 0.573473i \(-0.194404\pi\)
\(180\) 0 0
\(181\) 738.434i 0.303245i 0.988438 + 0.151622i \(0.0484498\pi\)
−0.988438 + 0.151622i \(0.951550\pi\)
\(182\) 184.989 52.8751i 0.0753424 0.0215350i
\(183\) 0 0
\(184\) −1333.05 −0.534096
\(185\) 2087.28 0.829514
\(186\) 0 0
\(187\) 23.4819i 0.00918270i
\(188\) 1029.54 0.399398
\(189\) 0 0
\(190\) −1044.98 −0.399005
\(191\) 3325.18i 1.25970i −0.776719 0.629848i \(-0.783117\pi\)
0.776719 0.629848i \(-0.216883\pi\)
\(192\) 0 0
\(193\) −3321.49 −1.23879 −0.619394 0.785080i \(-0.712622\pi\)
−0.619394 + 0.785080i \(0.712622\pi\)
\(194\) −1062.76 −0.393307
\(195\) 0 0
\(196\) 2176.12 1354.66i 0.793045 0.493681i
\(197\) 501.482i 0.181366i −0.995880 0.0906830i \(-0.971095\pi\)
0.995880 0.0906830i \(-0.0289050\pi\)
\(198\) 0 0
\(199\) 4067.59i 1.44896i −0.689294 0.724482i \(-0.742079\pi\)
0.689294 0.724482i \(-0.257921\pi\)
\(200\) 28.4787i 0.0100687i
\(201\) 0 0
\(202\) 1137.64i 0.396259i
\(203\) −1459.42 5105.94i −0.504587 1.76535i
\(204\) 0 0
\(205\) −1178.43 −0.401487
\(206\) −72.7943 −0.0246205
\(207\) 0 0
\(208\) 739.062i 0.246369i
\(209\) −32.9974 −0.0109210
\(210\) 0 0
\(211\) 4913.90 1.60326 0.801628 0.597823i \(-0.203968\pi\)
0.801628 + 0.597823i \(0.203968\pi\)
\(212\) 3133.37i 1.01510i
\(213\) 0 0
\(214\) −401.642 −0.128297
\(215\) 477.150 0.151355
\(216\) 0 0
\(217\) 1259.03 + 4404.87i 0.393865 + 1.37798i
\(218\) 290.754i 0.0903318i
\(219\) 0 0
\(220\) 20.9750i 0.00642789i
\(221\) 1325.20i 0.403360i
\(222\) 0 0
\(223\) 1193.16i 0.358295i −0.983822 0.179148i \(-0.942666\pi\)
0.983822 0.179148i \(-0.0573340\pi\)
\(224\) −648.027 2267.20i −0.193295 0.676265i
\(225\) 0 0
\(226\) −517.443 −0.152300
\(227\) −5608.63 −1.63990 −0.819951 0.572434i \(-0.805999\pi\)
−0.819951 + 0.572434i \(0.805999\pi\)
\(228\) 0 0
\(229\) 5333.05i 1.53894i 0.638681 + 0.769472i \(0.279480\pi\)
−0.638681 + 0.769472i \(0.720520\pi\)
\(230\) −953.388 −0.273324
\(231\) 0 0
\(232\) −3220.15 −0.911265
\(233\) 111.828i 0.0314424i 0.999876 + 0.0157212i \(0.00500442\pi\)
−0.999876 + 0.0157212i \(0.994996\pi\)
\(234\) 0 0
\(235\) 1524.54 0.423192
\(236\) 3251.75 0.896910
\(237\) 0 0
\(238\) 342.023 + 1196.61i 0.0931515 + 0.325901i
\(239\) 4771.99i 1.29152i −0.763539 0.645762i \(-0.776540\pi\)
0.763539 0.645762i \(-0.223460\pi\)
\(240\) 0 0
\(241\) 5336.30i 1.42631i 0.701005 + 0.713156i \(0.252735\pi\)
−0.701005 + 0.713156i \(0.747265\pi\)
\(242\) 965.987i 0.256595i
\(243\) 0 0
\(244\) 1410.93i 0.370186i
\(245\) 3222.39 2005.98i 0.840290 0.523092i
\(246\) 0 0
\(247\) −1862.21 −0.479715
\(248\) 2778.01 0.711305
\(249\) 0 0
\(250\) 1024.36i 0.259144i
\(251\) 3081.73 0.774970 0.387485 0.921876i \(-0.373344\pi\)
0.387485 + 0.921876i \(0.373344\pi\)
\(252\) 0 0
\(253\) −30.1052 −0.00748101
\(254\) 74.1542i 0.0183183i
\(255\) 0 0
\(256\) 1657.18 0.404585
\(257\) −2135.10 −0.518226 −0.259113 0.965847i \(-0.583430\pi\)
−0.259113 + 0.965847i \(0.583430\pi\)
\(258\) 0 0
\(259\) −3358.70 + 960.009i −0.805789 + 0.230317i
\(260\) 1183.72i 0.282352i
\(261\) 0 0
\(262\) 343.189i 0.0809248i
\(263\) 1650.23i 0.386911i −0.981109 0.193455i \(-0.938031\pi\)
0.981109 0.193455i \(-0.0619695\pi\)
\(264\) 0 0
\(265\) 4639.90i 1.07557i
\(266\) 1681.51 480.621i 0.387593 0.110785i
\(267\) 0 0
\(268\) 2770.32 0.631433
\(269\) 5477.73 1.24157 0.620787 0.783979i \(-0.286813\pi\)
0.620787 + 0.783979i \(0.286813\pi\)
\(270\) 0 0
\(271\) 3607.16i 0.808558i −0.914636 0.404279i \(-0.867523\pi\)
0.914636 0.404279i \(-0.132477\pi\)
\(272\) −4780.63 −1.06569
\(273\) 0 0
\(274\) −1901.15 −0.419171
\(275\) 0.643154i 0.000141032i
\(276\) 0 0
\(277\) 2977.98 0.645954 0.322977 0.946407i \(-0.395316\pi\)
0.322977 + 0.946407i \(0.395316\pi\)
\(278\) 901.707 0.194535
\(279\) 0 0
\(280\) −632.554 2213.06i −0.135008 0.472342i
\(281\) 5700.01i 1.21009i −0.796193 0.605043i \(-0.793156\pi\)
0.796193 0.605043i \(-0.206844\pi\)
\(282\) 0 0
\(283\) 550.467i 0.115625i −0.998327 0.0578125i \(-0.981587\pi\)
0.998327 0.0578125i \(-0.0184126\pi\)
\(284\) 195.522i 0.0408525i
\(285\) 0 0
\(286\) 2.63477i 0.000544746i
\(287\) 1896.23 541.996i 0.390004 0.111474i
\(288\) 0 0
\(289\) 3659.06 0.744771
\(290\) −2303.03 −0.466341
\(291\) 0 0
\(292\) 5444.58i 1.09116i
\(293\) −7830.00 −1.56121 −0.780603 0.625027i \(-0.785088\pi\)
−0.780603 + 0.625027i \(0.785088\pi\)
\(294\) 0 0
\(295\) 4815.19 0.950343
\(296\) 2118.22i 0.415943i
\(297\) 0 0
\(298\) 816.195 0.158661
\(299\) −1698.98 −0.328611
\(300\) 0 0
\(301\) −767.795 + 219.457i −0.147026 + 0.0420242i
\(302\) 345.186i 0.0657722i
\(303\) 0 0
\(304\) 6717.89i 1.26742i
\(305\) 2089.30i 0.392240i
\(306\) 0 0
\(307\) 5845.06i 1.08663i −0.839529 0.543315i \(-0.817169\pi\)
0.839529 0.543315i \(-0.182831\pi\)
\(308\) −9.64709 33.7514i −0.00178472 0.00624405i
\(309\) 0 0
\(310\) 1986.81 0.364011
\(311\) 5819.70 1.06111 0.530554 0.847651i \(-0.321984\pi\)
0.530554 + 0.847651i \(0.321984\pi\)
\(312\) 0 0
\(313\) 6644.41i 1.19989i −0.800043 0.599943i \(-0.795190\pi\)
0.800043 0.599943i \(-0.204810\pi\)
\(314\) 1403.19 0.252187
\(315\) 0 0
\(316\) 727.561 0.129521
\(317\) 6012.31i 1.06525i 0.846350 + 0.532626i \(0.178795\pi\)
−0.846350 + 0.532626i \(0.821205\pi\)
\(318\) 0 0
\(319\) −72.7230 −0.0127640
\(320\) 3548.65 0.619924
\(321\) 0 0
\(322\) 1534.12 438.494i 0.265507 0.0758892i
\(323\) 12045.7i 2.07505i
\(324\) 0 0
\(325\) 36.2964i 0.00619496i
\(326\) 539.500i 0.0916569i
\(327\) 0 0
\(328\) 1195.89i 0.201318i
\(329\) −2453.18 + 701.186i −0.411088 + 0.117500i
\(330\) 0 0
\(331\) −2742.49 −0.455410 −0.227705 0.973730i \(-0.573122\pi\)
−0.227705 + 0.973730i \(0.573122\pi\)
\(332\) −5998.89 −0.991662
\(333\) 0 0
\(334\) 2629.70i 0.430811i
\(335\) 4102.29 0.669050
\(336\) 0 0
\(337\) −3906.54 −0.631462 −0.315731 0.948849i \(-0.602250\pi\)
−0.315731 + 0.948849i \(0.602250\pi\)
\(338\) 1445.88i 0.232679i
\(339\) 0 0
\(340\) −7656.93 −1.22134
\(341\) 62.7377 0.00996316
\(342\) 0 0
\(343\) −4262.62 + 4709.96i −0.671019 + 0.741440i
\(344\) 484.223i 0.0758941i
\(345\) 0 0
\(346\) 2392.95i 0.371809i
\(347\) 588.749i 0.0910827i 0.998962 + 0.0455413i \(0.0145013\pi\)
−0.998962 + 0.0455413i \(0.985499\pi\)
\(348\) 0 0
\(349\) 7344.28i 1.12645i −0.826304 0.563224i \(-0.809561\pi\)
0.826304 0.563224i \(-0.190439\pi\)
\(350\) 9.36780 + 32.7743i 0.00143066 + 0.00500532i
\(351\) 0 0
\(352\) −32.2913 −0.00488957
\(353\) 8741.54 1.31803 0.659016 0.752129i \(-0.270973\pi\)
0.659016 + 0.752129i \(0.270973\pi\)
\(354\) 0 0
\(355\) 289.529i 0.0432863i
\(356\) 1768.30 0.263257
\(357\) 0 0
\(358\) 1993.59 0.294315
\(359\) 7817.81i 1.14933i 0.818390 + 0.574663i \(0.194867\pi\)
−0.818390 + 0.574663i \(0.805133\pi\)
\(360\) 0 0
\(361\) −10068.0 −1.46785
\(362\) −535.952 −0.0778149
\(363\) 0 0
\(364\) −544.433 1904.76i −0.0783957 0.274276i
\(365\) 8062.34i 1.15617i
\(366\) 0 0
\(367\) 2406.54i 0.342290i −0.985246 0.171145i \(-0.945253\pi\)
0.985246 0.171145i \(-0.0547466\pi\)
\(368\) 6129.05i 0.868204i
\(369\) 0 0
\(370\) 1514.94i 0.212859i
\(371\) −2134.04 7466.17i −0.298635 1.04481i
\(372\) 0 0
\(373\) 4370.16 0.606644 0.303322 0.952888i \(-0.401904\pi\)
0.303322 + 0.952888i \(0.401904\pi\)
\(374\) 17.0430 0.00235635
\(375\) 0 0
\(376\) 1547.14i 0.212201i
\(377\) −4104.12 −0.560671
\(378\) 0 0
\(379\) 11260.9 1.52620 0.763102 0.646278i \(-0.223675\pi\)
0.763102 + 0.646278i \(0.223675\pi\)
\(380\) 10759.8i 1.45254i
\(381\) 0 0
\(382\) 2413.40 0.323247
\(383\) −2973.14 −0.396659 −0.198330 0.980135i \(-0.563552\pi\)
−0.198330 + 0.980135i \(0.563552\pi\)
\(384\) 0 0
\(385\) −14.2854 49.9791i −0.00189105 0.00661603i
\(386\) 2410.72i 0.317882i
\(387\) 0 0
\(388\) 10942.8i 1.43179i
\(389\) 6598.27i 0.860014i −0.902826 0.430007i \(-0.858511\pi\)
0.902826 0.430007i \(-0.141489\pi\)
\(390\) 0 0
\(391\) 10989.9i 1.42144i
\(392\) 2035.72 + 3270.16i 0.262294 + 0.421347i
\(393\) 0 0
\(394\) 363.973 0.0465398
\(395\) 1077.37 0.137237
\(396\) 0 0
\(397\) 5908.48i 0.746947i −0.927641 0.373474i \(-0.878167\pi\)
0.927641 0.373474i \(-0.121833\pi\)
\(398\) 2952.24 0.371815
\(399\) 0 0
\(400\) −130.939 −0.0163673
\(401\) 5597.44i 0.697064i 0.937297 + 0.348532i \(0.113320\pi\)
−0.937297 + 0.348532i \(0.886680\pi\)
\(402\) 0 0
\(403\) 3540.60 0.437642
\(404\) −11713.9 −1.44254
\(405\) 0 0
\(406\) 3705.87 1059.24i 0.453003 0.129481i
\(407\) 47.8373i 0.00582606i
\(408\) 0 0
\(409\) 1095.31i 0.132420i 0.997806 + 0.0662099i \(0.0210907\pi\)
−0.997806 + 0.0662099i \(0.978909\pi\)
\(410\) 855.296i 0.103025i
\(411\) 0 0
\(412\) 749.533i 0.0896283i
\(413\) −7748.24 + 2214.66i −0.923162 + 0.263865i
\(414\) 0 0
\(415\) −8883.16 −1.05074
\(416\) −1822.36 −0.214780
\(417\) 0 0
\(418\) 23.9494i 0.00280240i
\(419\) 13593.6 1.58494 0.792468 0.609913i \(-0.208796\pi\)
0.792468 + 0.609913i \(0.208796\pi\)
\(420\) 0 0
\(421\) −5684.27 −0.658039 −0.329019 0.944323i \(-0.606718\pi\)
−0.329019 + 0.944323i \(0.606718\pi\)
\(422\) 3566.49i 0.411407i
\(423\) 0 0
\(424\) −4708.68 −0.539324
\(425\) 234.784 0.0267969
\(426\) 0 0
\(427\) 960.938 + 3361.95i 0.108906 + 0.381021i
\(428\) 4135.54i 0.467054i
\(429\) 0 0
\(430\) 346.314i 0.0388389i
\(431\) 4806.22i 0.537140i −0.963260 0.268570i \(-0.913449\pi\)
0.963260 0.268570i \(-0.0865511\pi\)
\(432\) 0 0
\(433\) 9621.49i 1.06785i 0.845532 + 0.533925i \(0.179284\pi\)
−0.845532 + 0.533925i \(0.820716\pi\)
\(434\) −3197.03 + 913.800i −0.353600 + 0.101069i
\(435\) 0 0
\(436\) −2993.77 −0.328844
\(437\) −15443.3 −1.69051
\(438\) 0 0
\(439\) 12932.6i 1.40601i 0.711183 + 0.703007i \(0.248160\pi\)
−0.711183 + 0.703007i \(0.751840\pi\)
\(440\) −31.5202 −0.00341516
\(441\) 0 0
\(442\) 961.823 0.103505
\(443\) 13164.7i 1.41191i 0.708259 + 0.705953i \(0.249481\pi\)
−0.708259 + 0.705953i \(0.750519\pi\)
\(444\) 0 0
\(445\) 2618.49 0.278940
\(446\) 865.990 0.0919413
\(447\) 0 0
\(448\) −5710.22 + 1632.14i −0.602194 + 0.172124i
\(449\) 7787.86i 0.818557i −0.912410 0.409278i \(-0.865781\pi\)
0.912410 0.409278i \(-0.134219\pi\)
\(450\) 0 0
\(451\) 27.0077i 0.00281983i
\(452\) 5327.90i 0.554432i
\(453\) 0 0
\(454\) 4070.72i 0.420811i
\(455\) −806.197 2820.57i −0.0830662 0.290616i
\(456\) 0 0
\(457\) 4817.13 0.493076 0.246538 0.969133i \(-0.420707\pi\)
0.246538 + 0.969133i \(0.420707\pi\)
\(458\) −3870.71 −0.394904
\(459\) 0 0
\(460\) 9816.65i 0.995008i
\(461\) 51.2892 0.00518172 0.00259086 0.999997i \(-0.499175\pi\)
0.00259086 + 0.999997i \(0.499175\pi\)
\(462\) 0 0
\(463\) −9574.86 −0.961083 −0.480542 0.876972i \(-0.659560\pi\)
−0.480542 + 0.876972i \(0.659560\pi\)
\(464\) 14805.5i 1.48131i
\(465\) 0 0
\(466\) −81.1641 −0.00806835
\(467\) 12806.5 1.26898 0.634492 0.772929i \(-0.281209\pi\)
0.634492 + 0.772929i \(0.281209\pi\)
\(468\) 0 0
\(469\) −6601.09 + 1886.77i −0.649915 + 0.185764i
\(470\) 1106.51i 0.108594i
\(471\) 0 0
\(472\) 4886.57i 0.476531i
\(473\) 10.9356i 0.00106304i
\(474\) 0 0
\(475\) 329.925i 0.0318695i
\(476\) 12321.0 3521.67i 1.18641 0.339108i
\(477\) 0 0
\(478\) 3463.49 0.331415
\(479\) 5085.57 0.485106 0.242553 0.970138i \(-0.422015\pi\)
0.242553 + 0.970138i \(0.422015\pi\)
\(480\) 0 0
\(481\) 2699.70i 0.255916i
\(482\) −3873.06 −0.366002
\(483\) 0 0
\(484\) 9946.38 0.934107
\(485\) 16204.1i 1.51709i
\(486\) 0 0
\(487\) 16991.3 1.58100 0.790501 0.612461i \(-0.209820\pi\)
0.790501 + 0.612461i \(0.209820\pi\)
\(488\) 2120.27 0.196681
\(489\) 0 0
\(490\) 1455.93 + 2338.80i 0.134229 + 0.215625i
\(491\) 9301.80i 0.854958i −0.904025 0.427479i \(-0.859402\pi\)
0.904025 0.427479i \(-0.140598\pi\)
\(492\) 0 0
\(493\) 26547.5i 2.42524i
\(494\) 1351.58i 0.123098i
\(495\) 0 0
\(496\) 12772.6i 1.15627i
\(497\) −133.164 465.889i −0.0120185 0.0420482i
\(498\) 0 0
\(499\) −500.665 −0.0449155 −0.0224577 0.999748i \(-0.507149\pi\)
−0.0224577 + 0.999748i \(0.507149\pi\)
\(500\) −10547.4 −0.943386
\(501\) 0 0
\(502\) 2236.71i 0.198863i
\(503\) −3581.09 −0.317441 −0.158721 0.987324i \(-0.550737\pi\)
−0.158721 + 0.987324i \(0.550737\pi\)
\(504\) 0 0
\(505\) −17345.9 −1.52848
\(506\) 21.8502i 0.00191968i
\(507\) 0 0
\(508\) 763.536 0.0666859
\(509\) 10170.7 0.885679 0.442839 0.896601i \(-0.353971\pi\)
0.442839 + 0.896601i \(0.353971\pi\)
\(510\) 0 0
\(511\) 3708.13 + 12973.3i 0.321014 + 1.12310i
\(512\) 11213.1i 0.967882i
\(513\) 0 0
\(514\) 1549.65i 0.132981i
\(515\) 1109.91i 0.0949679i
\(516\) 0 0
\(517\) 34.9402i 0.00297228i
\(518\) −696.770 2437.73i −0.0591010 0.206771i
\(519\) 0 0
\(520\) −1778.84 −0.150014
\(521\) −389.899 −0.0327865 −0.0163933 0.999866i \(-0.505218\pi\)
−0.0163933 + 0.999866i \(0.505218\pi\)
\(522\) 0 0
\(523\) 14306.4i 1.19613i −0.801448 0.598065i \(-0.795937\pi\)
0.801448 0.598065i \(-0.204063\pi\)
\(524\) −3533.68 −0.294598
\(525\) 0 0
\(526\) 1197.73 0.0992842
\(527\) 22902.4i 1.89306i
\(528\) 0 0
\(529\) −1922.70 −0.158026
\(530\) −3367.62 −0.276000
\(531\) 0 0
\(532\) −4948.76 17313.8i −0.403301 1.41099i
\(533\) 1524.18i 0.123864i
\(534\) 0 0
\(535\) 6123.91i 0.494878i
\(536\) 4163.10i 0.335482i
\(537\) 0 0
\(538\) 3975.71i 0.318597i
\(539\) 45.9740 + 73.8523i 0.00367392 + 0.00590175i
\(540\) 0 0
\(541\) 21775.6 1.73051 0.865257 0.501328i \(-0.167155\pi\)
0.865257 + 0.501328i \(0.167155\pi\)
\(542\) 2618.06 0.207482
\(543\) 0 0
\(544\) 11787.9i 0.929050i
\(545\) −4433.18 −0.348434
\(546\) 0 0
\(547\) −17921.7 −1.40087 −0.700436 0.713715i \(-0.747011\pi\)
−0.700436 + 0.713715i \(0.747011\pi\)
\(548\) 19575.4i 1.52595i
\(549\) 0 0
\(550\) 0.466799 3.61897e−5
\(551\) −37305.4 −2.88433
\(552\) 0 0
\(553\) −1733.63 + 495.518i −0.133312 + 0.0381041i
\(554\) 2161.40i 0.165757i
\(555\) 0 0
\(556\) 9284.51i 0.708185i
\(557\) 19309.8i 1.46891i 0.678657 + 0.734455i \(0.262562\pi\)
−0.678657 + 0.734455i \(0.737438\pi\)
\(558\) 0 0
\(559\) 617.148i 0.0466951i
\(560\) −10175.2 + 2908.34i −0.767819 + 0.219464i
\(561\) 0 0
\(562\) 4137.04 0.310517
\(563\) −6464.49 −0.483918 −0.241959 0.970287i \(-0.577790\pi\)
−0.241959 + 0.970287i \(0.577790\pi\)
\(564\) 0 0
\(565\) 7889.55i 0.587462i
\(566\) 399.527 0.0296702
\(567\) 0 0
\(568\) −293.821 −0.0217050
\(569\) 19961.2i 1.47068i 0.677698 + 0.735341i \(0.262978\pi\)
−0.677698 + 0.735341i \(0.737022\pi\)
\(570\) 0 0
\(571\) −12515.1 −0.917235 −0.458617 0.888634i \(-0.651655\pi\)
−0.458617 + 0.888634i \(0.651655\pi\)
\(572\) −27.1292 −0.00198309
\(573\) 0 0
\(574\) 393.378 + 1376.28i 0.0286051 + 0.100078i
\(575\) 301.007i 0.0218310i
\(576\) 0 0
\(577\) 547.118i 0.0394746i 0.999805 + 0.0197373i \(0.00628298\pi\)
−0.999805 + 0.0197373i \(0.993717\pi\)
\(578\) 2655.73i 0.191114i
\(579\) 0 0
\(580\) 23713.4i 1.69766i
\(581\) 14294.1 4085.65i 1.02069 0.291741i
\(582\) 0 0
\(583\) −106.339 −0.00755425
\(584\) 8181.85 0.579739
\(585\) 0 0
\(586\) 5682.98i 0.400617i
\(587\) −1804.83 −0.126905 −0.0634525 0.997985i \(-0.520211\pi\)
−0.0634525 + 0.997985i \(0.520211\pi\)
\(588\) 0 0
\(589\) 32183.2 2.25142
\(590\) 3494.84i 0.243865i
\(591\) 0 0
\(592\) 9739.11 0.676140
\(593\) 20359.7 1.40990 0.704951 0.709256i \(-0.250969\pi\)
0.704951 + 0.709256i \(0.250969\pi\)
\(594\) 0 0
\(595\) 18244.9 5214.89i 1.25709 0.359311i
\(596\) 8404.03i 0.577588i
\(597\) 0 0
\(598\) 1233.11i 0.0843241i
\(599\) 4317.81i 0.294526i 0.989097 + 0.147263i \(0.0470463\pi\)
−0.989097 + 0.147263i \(0.952954\pi\)
\(600\) 0 0
\(601\) 7999.97i 0.542971i 0.962443 + 0.271485i \(0.0875149\pi\)
−0.962443 + 0.271485i \(0.912485\pi\)
\(602\) −159.281 557.262i −0.0107837 0.0377281i
\(603\) 0 0
\(604\) −3554.24 −0.239437
\(605\) 14728.6 0.989757
\(606\) 0 0
\(607\) 3556.10i 0.237789i 0.992907 + 0.118894i \(0.0379350\pi\)
−0.992907 + 0.118894i \(0.962065\pi\)
\(608\) −16564.7 −1.10492
\(609\) 0 0
\(610\) 1516.41 0.100652
\(611\) 1971.85i 0.130560i
\(612\) 0 0
\(613\) −7113.75 −0.468714 −0.234357 0.972151i \(-0.575298\pi\)
−0.234357 + 0.972151i \(0.575298\pi\)
\(614\) 4242.32 0.278837
\(615\) 0 0
\(616\) 50.7200 14.4972i 0.00331748 0.000948227i
\(617\) 913.301i 0.0595917i −0.999556 0.0297959i \(-0.990514\pi\)
0.999556 0.0297959i \(-0.00948572\pi\)
\(618\) 0 0
\(619\) 946.665i 0.0614696i −0.999528 0.0307348i \(-0.990215\pi\)
0.999528 0.0307348i \(-0.00978473\pi\)
\(620\) 20457.4i 1.32514i
\(621\) 0 0
\(622\) 4223.91i 0.272288i
\(623\) −4213.48 + 1204.33i −0.270962 + 0.0774485i
\(624\) 0 0
\(625\) −15301.6 −0.979302
\(626\) 4822.48 0.307900
\(627\) 0 0
\(628\) 14448.1i 0.918062i
\(629\) −17463.0 −1.10699
\(630\) 0 0
\(631\) 3209.46 0.202482 0.101241 0.994862i \(-0.467719\pi\)
0.101241 + 0.994862i \(0.467719\pi\)
\(632\) 1093.34i 0.0688146i
\(633\) 0 0
\(634\) −4363.71 −0.273352
\(635\) 1130.64 0.0706587
\(636\) 0 0
\(637\) 2594.54 + 4167.85i 0.161381 + 0.259241i
\(638\) 52.7820i 0.00327533i
\(639\) 0 0
\(640\) 13847.3i 0.855253i
\(641\) 14303.4i 0.881359i 0.897665 + 0.440679i \(0.145262\pi\)
−0.897665 + 0.440679i \(0.854738\pi\)
\(642\) 0 0
\(643\) 974.886i 0.0597912i 0.999553 + 0.0298956i \(0.00951748\pi\)
−0.999553 + 0.0298956i \(0.990483\pi\)
\(644\) −4514.99 15796.2i −0.276267 0.966549i
\(645\) 0 0
\(646\) 8742.73 0.532474
\(647\) −12668.5 −0.769783 −0.384891 0.922962i \(-0.625761\pi\)
−0.384891 + 0.922962i \(0.625761\pi\)
\(648\) 0 0
\(649\) 110.357i 0.00667470i
\(650\) 26.3438 0.00158967
\(651\) 0 0
\(652\) −5555.01 −0.333667
\(653\) 15873.3i 0.951256i 0.879646 + 0.475628i \(0.157779\pi\)
−0.879646 + 0.475628i \(0.842221\pi\)
\(654\) 0 0
\(655\) −5232.67 −0.312149
\(656\) −5498.45 −0.327254
\(657\) 0 0
\(658\) −508.918 1780.50i −0.0301515 0.105488i
\(659\) 30496.7i 1.80271i −0.433084 0.901353i \(-0.642575\pi\)
0.433084 0.901353i \(-0.357425\pi\)
\(660\) 0 0
\(661\) 16702.5i 0.982829i −0.870926 0.491415i \(-0.836480\pi\)
0.870926 0.491415i \(-0.163520\pi\)
\(662\) 1990.49i 0.116862i
\(663\) 0 0
\(664\) 9014.83i 0.526873i
\(665\) −7328.13 25638.3i −0.427327 1.49505i
\(666\) 0 0
\(667\) −34035.5 −1.97580
\(668\) 27076.9 1.56832
\(669\) 0 0
\(670\) 2977.42i 0.171683i
\(671\) 47.8836 0.00275488
\(672\) 0 0
\(673\) 24890.4 1.42564 0.712820 0.701347i \(-0.247418\pi\)
0.712820 + 0.701347i \(0.247418\pi\)
\(674\) 2835.35i 0.162038i
\(675\) 0 0
\(676\) 14887.6 0.847043
\(677\) 1512.55 0.0858672 0.0429336 0.999078i \(-0.486330\pi\)
0.0429336 + 0.999078i \(0.486330\pi\)
\(678\) 0 0
\(679\) −7452.79 26074.4i −0.421225 1.47370i
\(680\) 11506.5i 0.648901i
\(681\) 0 0
\(682\) 45.5347i 0.00255662i
\(683\) 2180.44i 0.122156i −0.998133 0.0610778i \(-0.980546\pi\)
0.998133 0.0610778i \(-0.0194538\pi\)
\(684\) 0 0
\(685\) 28987.3i 1.61686i
\(686\) −3418.47 3093.79i −0.190259 0.172189i
\(687\) 0 0
\(688\) 2226.35 0.123370
\(689\) −6001.26 −0.331828
\(690\) 0 0
\(691\) 27904.8i 1.53625i 0.640301 + 0.768124i \(0.278810\pi\)
−0.640301 + 0.768124i \(0.721190\pi\)
\(692\) 24639.2 1.35353
\(693\) 0 0
\(694\) −427.311 −0.0233725
\(695\) 13748.5i 0.750375i
\(696\) 0 0
\(697\) 9859.17 0.535786
\(698\) 5330.44 0.289055
\(699\) 0 0
\(700\) 337.464 96.4565i 0.0182213 0.00520816i
\(701\) 15696.6i 0.845722i −0.906195 0.422861i \(-0.861026\pi\)
0.906195 0.422861i \(-0.138974\pi\)
\(702\) 0 0
\(703\) 24539.6i 1.31654i
\(704\) 81.3297i 0.00435402i
\(705\) 0 0
\(706\) 6344.57i 0.338217i
\(707\) 27911.7 7977.94i 1.48476 0.424386i
\(708\) 0 0
\(709\) −18506.9 −0.980313 −0.490157 0.871634i \(-0.663060\pi\)
−0.490157 + 0.871634i \(0.663060\pi\)
\(710\) −210.139 −0.0111076
\(711\) 0 0
\(712\) 2657.31i 0.139869i
\(713\) 29362.3 1.54225
\(714\) 0 0
\(715\) −40.1729 −0.00210123
\(716\) 20527.2i 1.07142i
\(717\) 0 0
\(718\) −5674.13 −0.294926
\(719\) −12998.8 −0.674232 −0.337116 0.941463i \(-0.609451\pi\)
−0.337116 + 0.941463i \(0.609451\pi\)
\(720\) 0 0
\(721\) −510.483 1785.98i −0.0263681 0.0922517i
\(722\) 7307.32i 0.376662i
\(723\) 0 0
\(724\) 5518.48i 0.283277i
\(725\) 727.121i 0.0372477i
\(726\) 0 0
\(727\) 1900.80i 0.0969693i −0.998824 0.0484846i \(-0.984561\pi\)
0.998824 0.0484846i \(-0.0154392\pi\)
\(728\) 2862.38 818.148i 0.145724 0.0416519i
\(729\) 0 0
\(730\) 5851.61 0.296682
\(731\) −3992.03 −0.201984
\(732\) 0 0
\(733\) 5547.48i 0.279538i 0.990184 + 0.139769i \(0.0446359\pi\)
−0.990184 + 0.139769i \(0.955364\pi\)
\(734\) 1746.65 0.0878341
\(735\) 0 0
\(736\) −15112.8 −0.756883
\(737\) 94.0181i 0.00469905i
\(738\) 0 0
\(739\) −15681.5 −0.780586 −0.390293 0.920691i \(-0.627626\pi\)
−0.390293 + 0.920691i \(0.627626\pi\)
\(740\) 15598.7 0.774892
\(741\) 0 0
\(742\) 5418.91 1548.88i 0.268106 0.0766321i
\(743\) 18081.9i 0.892815i −0.894830 0.446408i \(-0.852703\pi\)
0.894830 0.446408i \(-0.147297\pi\)
\(744\) 0 0
\(745\) 12444.7i 0.611998i
\(746\) 3171.84i 0.155669i
\(747\) 0 0
\(748\) 175.485i 0.00857804i
\(749\) −2816.59 9854.14i −0.137404 0.480724i
\(750\) 0 0
\(751\) 22182.7 1.07784 0.538921 0.842357i \(-0.318832\pi\)
0.538921 + 0.842357i \(0.318832\pi\)
\(752\) 7113.40 0.344945
\(753\) 0 0
\(754\) 2978.75i 0.143872i
\(755\) −5263.12 −0.253701
\(756\) 0 0
\(757\) −2531.12 −0.121526 −0.0607630 0.998152i \(-0.519353\pi\)
−0.0607630 + 0.998152i \(0.519353\pi\)
\(758\) 8173.08i 0.391636i
\(759\) 0 0
\(760\) −16169.2 −0.771737
\(761\) 3794.03 0.180727 0.0903637 0.995909i \(-0.471197\pi\)
0.0903637 + 0.995909i \(0.471197\pi\)
\(762\) 0 0
\(763\) 7133.54 2038.96i 0.338469 0.0967437i
\(764\) 24849.8i 1.17675i
\(765\) 0 0
\(766\) 2157.89i 0.101786i
\(767\) 6227.98i 0.293193i
\(768\) 0 0
\(769\) 6378.70i 0.299118i 0.988753 + 0.149559i \(0.0477854\pi\)
−0.988753 + 0.149559i \(0.952215\pi\)
\(770\) 36.2746 10.3683i 0.00169772 0.000485256i
\(771\) 0 0
\(772\) −24822.2 −1.15722
\(773\) −29728.9 −1.38328 −0.691638 0.722244i \(-0.743111\pi\)
−0.691638 + 0.722244i \(0.743111\pi\)
\(774\) 0 0
\(775\) 627.283i 0.0290744i
\(776\) −16444.3 −0.760716
\(777\) 0 0
\(778\) 4788.99 0.220686
\(779\) 13854.4i 0.637209i
\(780\) 0 0
\(781\) −6.63557 −0.000304020
\(782\) 7976.42 0.364752
\(783\) 0 0
\(784\) 15035.5 9359.77i 0.684924 0.426374i
\(785\) 21394.8i 0.972755i
\(786\) 0 0
\(787\) 3738.07i 0.169311i −0.996410 0.0846556i \(-0.973021\pi\)
0.996410 0.0846556i \(-0.0269790\pi\)
\(788\) 3747.68i 0.169423i
\(789\) 0 0
\(790\) 781.952i 0.0352160i
\(791\) −3628.66 12695.3i −0.163110 0.570660i
\(792\) 0 0
\(793\) 2702.31 0.121011
\(794\) 4288.35 0.191672
\(795\) 0 0
\(796\) 30398.0i 1.35355i
\(797\) 10754.8 0.477985 0.238992 0.971021i \(-0.423183\pi\)
0.238992 + 0.971021i \(0.423183\pi\)
\(798\) 0 0
\(799\) −12754.9 −0.564751
\(800\) 322.864i 0.0142687i
\(801\) 0 0
\(802\) −4062.60 −0.178872
\(803\) 184.776 0.00812032
\(804\) 0 0
\(805\) −6685.81 23391.0i −0.292725 1.02413i
\(806\) 2569.75i 0.112302i
\(807\) 0 0
\(808\) 17603.0i 0.766426i
\(809\) 31037.7i 1.34886i 0.738340 + 0.674429i \(0.235610\pi\)
−0.738340 + 0.674429i \(0.764390\pi\)
\(810\) 0 0
\(811\) 3943.22i 0.170734i −0.996350 0.0853669i \(-0.972794\pi\)
0.996350 0.0853669i \(-0.0272062\pi\)
\(812\) −10906.6 38157.8i −0.471361 1.64911i
\(813\) 0 0
\(814\) −34.7201 −0.00149501
\(815\) −8225.87 −0.353546
\(816\) 0 0
\(817\) 5609.72i 0.240219i
\(818\) −794.973 −0.0339799
\(819\) 0 0
\(820\) −8806.64 −0.375050
\(821\) 28968.6i 1.23144i 0.787965 + 0.615720i \(0.211135\pi\)
−0.787965 + 0.615720i \(0.788865\pi\)
\(822\) 0 0
\(823\) 37286.3 1.57924 0.789622 0.613594i \(-0.210277\pi\)
0.789622 + 0.613594i \(0.210277\pi\)
\(824\) −1126.36 −0.0476198
\(825\) 0 0
\(826\) −1607.39 5623.64i −0.0677098 0.236890i
\(827\) 13151.6i 0.552992i −0.961015 0.276496i \(-0.910827\pi\)
0.961015 0.276496i \(-0.0891732\pi\)
\(828\) 0 0
\(829\) 27671.1i 1.15929i −0.814867 0.579647i \(-0.803190\pi\)
0.814867 0.579647i \(-0.196810\pi\)
\(830\) 6447.35i 0.269628i
\(831\) 0 0
\(832\) 4589.84i 0.191255i
\(833\) −26959.8 + 16782.8i −1.12137 + 0.698068i
\(834\) 0 0
\(835\) 40095.6 1.66175
\(836\) −246.597 −0.0102018
\(837\) 0 0
\(838\) 9866.14i 0.406707i
\(839\) 16997.4 0.699424 0.349712 0.936857i \(-0.386279\pi\)
0.349712 + 0.936857i \(0.386279\pi\)
\(840\) 0 0
\(841\) −57828.3 −2.37108
\(842\) 4125.62i 0.168858i
\(843\) 0 0
\(844\) 36722.7 1.49769
\(845\) 22045.6 0.897506
\(846\) 0 0
\(847\) −23700.2 + 6774.16i −0.961449 + 0.274809i
\(848\) 21649.4i 0.876703i
\(849\) 0 0
\(850\) 170.405i 0.00687628i
\(851\) 22388.6i 0.901848i
\(852\) 0 0
\(853\) 15161.5i 0.608582i −0.952579 0.304291i \(-0.901581\pi\)
0.952579 0.304291i \(-0.0984195\pi\)
\(854\) −2440.09 + 697.445i −0.0977729 + 0.0279462i
\(855\) 0 0
\(856\) −6214.69 −0.248147
\(857\) −28373.7 −1.13095 −0.565477 0.824764i \(-0.691308\pi\)
−0.565477 + 0.824764i \(0.691308\pi\)
\(858\) 0 0
\(859\) 38580.5i 1.53242i 0.642590 + 0.766210i \(0.277860\pi\)
−0.642590 + 0.766210i \(0.722140\pi\)
\(860\) 3565.85 0.141389
\(861\) 0 0
\(862\) 3488.33 0.137834
\(863\) 112.649i 0.00444337i 0.999998 + 0.00222168i \(0.000707185\pi\)
−0.999998 + 0.00222168i \(0.999293\pi\)
\(864\) 0 0
\(865\) 36485.8 1.43417
\(866\) −6983.23 −0.274018
\(867\) 0 0
\(868\) 9409.03 + 32918.5i 0.367930 + 1.28724i
\(869\) 24.6917i 0.000963878i
\(870\) 0 0
\(871\) 5305.91i 0.206411i
\(872\) 4498.90i 0.174716i
\(873\) 0 0
\(874\) 11208.7i 0.433799i
\(875\) 25132.2 7183.49i 0.970999 0.277538i
\(876\) 0 0
\(877\) −20758.6 −0.799279 −0.399639 0.916672i \(-0.630865\pi\)
−0.399639 + 0.916672i \(0.630865\pi\)
\(878\) −9386.43 −0.360794
\(879\) 0 0
\(880\) 144.923i 0.00555154i
\(881\) 18603.4 0.711425 0.355712 0.934595i \(-0.384238\pi\)
0.355712 + 0.934595i \(0.384238\pi\)
\(882\) 0 0
\(883\) 17188.8 0.655096 0.327548 0.944835i \(-0.393778\pi\)
0.327548 + 0.944835i \(0.393778\pi\)
\(884\) 9903.50i 0.376800i
\(885\) 0 0
\(886\) −9554.89 −0.362305
\(887\) 13230.6 0.500835 0.250417 0.968138i \(-0.419432\pi\)
0.250417 + 0.968138i \(0.419432\pi\)
\(888\) 0 0
\(889\) −1819.35 + 520.020i −0.0686377 + 0.0196186i
\(890\) 1900.49i 0.0715782i
\(891\) 0 0
\(892\) 8916.74i 0.334703i
\(893\) 17923.6i 0.671657i
\(894\) 0 0
\(895\) 30396.7i 1.13525i
\(896\) −6368.82 22282.0i −0.237463 0.830792i
\(897\) 0 0
\(898\) 5652.39 0.210048
\(899\) 70928.4 2.63136
\(900\) 0 0
\(901\) 38819.2i 1.43535i
\(902\) 19.6021 0.000723590
\(903\) 0 0
\(904\) −8006.50 −0.294571
\(905\) 8171.77i 0.300153i
\(906\) 0 0
\(907\) −43505.9 −1.59271 −0.796356 0.604828i \(-0.793242\pi\)
−0.796356 + 0.604828i \(0.793242\pi\)
\(908\) −41914.5 −1.53192
\(909\) 0 0
\(910\) 2047.16 585.134i 0.0745743 0.0213154i
\(911\) 17204.7i 0.625703i 0.949802 + 0.312852i \(0.101284\pi\)
−0.949802 + 0.312852i \(0.898716\pi\)
\(912\) 0 0
\(913\) 203.588i 0.00737984i
\(914\) 3496.25i 0.126527i
\(915\) 0 0
\(916\) 39855.1i 1.43761i
\(917\) 8420.02 2406.68i 0.303221 0.0866690i
\(918\) 0 0
\(919\) 18441.8 0.661956 0.330978 0.943639i \(-0.392621\pi\)
0.330978 + 0.943639i \(0.392621\pi\)
\(920\) −14752.0 −0.528650
\(921\) 0 0
\(922\) 37.2254i 0.00132967i
\(923\) −374.478 −0.0133544
\(924\) 0 0
\(925\) −478.302 −0.0170016
\(926\) 6949.39i 0.246621i
\(927\) 0 0
\(928\) −36507.0 −1.29138
\(929\) −34469.4 −1.21734 −0.608668 0.793425i \(-0.708296\pi\)
−0.608668 + 0.793425i \(0.708296\pi\)
\(930\) 0 0
\(931\) 23583.7 + 37884.7i 0.830210 + 1.33364i
\(932\) 835.713i 0.0293720i
\(933\) 0 0
\(934\) 9294.92i 0.325631i
\(935\) 259.859i 0.00908908i
\(936\) 0 0
\(937\) 8204.91i 0.286065i −0.989718 0.143032i \(-0.954315\pi\)
0.989718 0.143032i \(-0.0456853\pi\)
\(938\) −1369.41 4791.04i −0.0476683 0.166773i
\(939\) 0 0
\(940\) 11393.2 0.395326
\(941\) 23911.3 0.828359 0.414179 0.910195i \(-0.364069\pi\)
0.414179 + 0.910195i \(0.364069\pi\)
\(942\) 0 0
\(943\) 12640.1i 0.436497i
\(944\) 22467.3 0.774628
\(945\) 0 0
\(946\) −7.93698 −0.000272784
\(947\) 39859.6i 1.36775i −0.729597 0.683877i \(-0.760292\pi\)
0.729597 0.683877i \(-0.239708\pi\)
\(948\) 0 0
\(949\) 10427.9 0.356694
\(950\) 239.458 0.00817794
\(951\) 0 0
\(952\) 5292.20 + 18515.3i 0.180169 + 0.630342i
\(953\) 10983.4i 0.373334i 0.982423 + 0.186667i \(0.0597685\pi\)
−0.982423 + 0.186667i \(0.940232\pi\)
\(954\) 0 0
\(955\) 36797.6i 1.24685i
\(956\) 35662.1i 1.20648i
\(957\) 0 0
\(958\) 3691.09i 0.124482i
\(959\) −13332.2 46644.2i −0.448925 1.57061i
\(960\) 0 0
\(961\) −31398.5 −1.05396
\(962\) −1959.43 −0.0656699
\(963\) 0 0
\(964\) 39879.3i 1.33239i
\(965\) −36756.8 −1.22616
\(966\) 0 0
\(967\) 14827.6 0.493096 0.246548 0.969131i \(-0.420704\pi\)
0.246548 + 0.969131i \(0.420704\pi\)
\(968\) 14946.9i 0.496294i
\(969\) 0 0
\(970\) −11760.9 −0.389297
\(971\) 8249.56 0.272648 0.136324 0.990664i \(-0.456471\pi\)
0.136324 + 0.990664i \(0.456471\pi\)
\(972\) 0 0
\(973\) 6323.38 + 22123.1i 0.208344 + 0.728913i
\(974\) 12332.2i 0.405697i
\(975\) 0 0
\(976\) 9748.54i 0.319716i
\(977\) 4774.55i 0.156347i −0.996940 0.0781737i \(-0.975091\pi\)
0.996940 0.0781737i \(-0.0249089\pi\)
\(978\) 0 0
\(979\) 60.0119i 0.00195913i
\(980\) 24081.7 14991.1i 0.784960 0.488648i
\(981\) 0 0
\(982\) 6751.20 0.219389
\(983\) 17940.0 0.582094 0.291047 0.956709i \(-0.405996\pi\)
0.291047 + 0.956709i \(0.405996\pi\)
\(984\) 0 0
\(985\) 5549.57i 0.179517i
\(986\) 19268.1 0.622333
\(987\) 0 0
\(988\) −13916.7 −0.448127
\(989\) 5118.02i 0.164554i
\(990\) 0 0
\(991\) 1714.50 0.0549574 0.0274787 0.999622i \(-0.491252\pi\)
0.0274787 + 0.999622i \(0.491252\pi\)
\(992\) 31494.4 1.00801
\(993\) 0 0
\(994\) 338.140 96.6498i 0.0107899 0.00308405i
\(995\) 45013.4i 1.43419i
\(996\) 0 0
\(997\) 34637.3i 1.10028i 0.835074 + 0.550138i \(0.185425\pi\)
−0.835074 + 0.550138i \(0.814575\pi\)
\(998\) 363.380i 0.0115256i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 567.4.c.c.566.12 44
3.2 odd 2 inner 567.4.c.c.566.33 44
7.6 odd 2 inner 567.4.c.c.566.34 44
9.2 odd 6 189.4.o.a.125.10 44
9.4 even 3 189.4.o.a.62.9 44
9.5 odd 6 63.4.o.a.20.14 yes 44
9.7 even 3 63.4.o.a.41.13 yes 44
21.20 even 2 inner 567.4.c.c.566.11 44
63.13 odd 6 189.4.o.a.62.10 44
63.20 even 6 189.4.o.a.125.9 44
63.34 odd 6 63.4.o.a.41.14 yes 44
63.41 even 6 63.4.o.a.20.13 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.o.a.20.13 44 63.41 even 6
63.4.o.a.20.14 yes 44 9.5 odd 6
63.4.o.a.41.13 yes 44 9.7 even 3
63.4.o.a.41.14 yes 44 63.34 odd 6
189.4.o.a.62.9 44 9.4 even 3
189.4.o.a.62.10 44 63.13 odd 6
189.4.o.a.125.9 44 63.20 even 6
189.4.o.a.125.10 44 9.2 odd 6
567.4.c.c.566.11 44 21.20 even 2 inner
567.4.c.c.566.12 44 1.1 even 1 trivial
567.4.c.c.566.33 44 3.2 odd 2 inner
567.4.c.c.566.34 44 7.6 odd 2 inner