Properties

Label 1344.4.b.i.895.5
Level $1344$
Weight $4$
Character 1344.895
Analytic conductor $79.299$
Analytic rank $0$
Dimension $24$
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] = [1344,4,Mod(895,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 895.5
Character \(\chi\) \(=\) 1344.895
Dual form 1344.4.b.i.895.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -13.1053i q^{5} +(9.87442 - 15.6683i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -13.1053i q^{5} +(9.87442 - 15.6683i) q^{7} +9.00000 q^{9} +66.5093i q^{11} +33.0750i q^{13} +39.3160i q^{15} +86.2229i q^{17} +21.0063 q^{19} +(-29.6233 + 47.0049i) q^{21} -218.669i q^{23} -46.7501 q^{25} -27.0000 q^{27} +133.471 q^{29} -324.428 q^{31} -199.528i q^{33} +(-205.339 - 129.408i) q^{35} +73.1277 q^{37} -99.2251i q^{39} -309.181i q^{41} -28.7269i q^{43} -117.948i q^{45} -162.730 q^{47} +(-147.992 - 309.431i) q^{49} -258.669i q^{51} -575.828 q^{53} +871.627 q^{55} -63.0188 q^{57} -861.347 q^{59} -367.037i q^{61} +(88.8698 - 141.015i) q^{63} +433.460 q^{65} +818.701i q^{67} +656.006i q^{69} -407.234i q^{71} -515.657i q^{73} +140.250 q^{75} +(1042.09 + 656.740i) q^{77} +1135.55i q^{79} +81.0000 q^{81} -941.158 q^{83} +1129.98 q^{85} -400.412 q^{87} +628.192i q^{89} +(518.230 + 326.597i) q^{91} +973.285 q^{93} -275.294i q^{95} -522.331i q^{97} +598.583i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9} + 56 q^{19} - 60 q^{21} - 432 q^{25} - 648 q^{27} - 464 q^{31} - 568 q^{35} - 504 q^{37} - 560 q^{47} - 128 q^{49} + 784 q^{53} - 424 q^{55} - 168 q^{57} - 800 q^{59} + 180 q^{63} + 560 q^{65} + 1296 q^{75} + 1568 q^{77} + 1944 q^{81} - 1936 q^{83} - 3000 q^{85} + 496 q^{91} + 1392 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 13.1053i 1.17218i −0.810247 0.586089i \(-0.800667\pi\)
0.810247 0.586089i \(-0.199333\pi\)
\(6\) 0 0
\(7\) 9.87442 15.6683i 0.533169 0.846009i
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 66.5093i 1.82303i 0.411270 + 0.911513i \(0.365085\pi\)
−0.411270 + 0.911513i \(0.634915\pi\)
\(12\) 0 0
\(13\) 33.0750i 0.705643i 0.935691 + 0.352822i \(0.114778\pi\)
−0.935691 + 0.352822i \(0.885222\pi\)
\(14\) 0 0
\(15\) 39.3160i 0.676757i
\(16\) 0 0
\(17\) 86.2229i 1.23013i 0.788478 + 0.615063i \(0.210869\pi\)
−0.788478 + 0.615063i \(0.789131\pi\)
\(18\) 0 0
\(19\) 21.0063 0.253640 0.126820 0.991926i \(-0.459523\pi\)
0.126820 + 0.991926i \(0.459523\pi\)
\(20\) 0 0
\(21\) −29.6233 + 47.0049i −0.307825 + 0.488444i
\(22\) 0 0
\(23\) 218.669i 1.98242i −0.132307 0.991209i \(-0.542239\pi\)
0.132307 0.991209i \(-0.457761\pi\)
\(24\) 0 0
\(25\) −46.7501 −0.374001
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 133.471 0.854652 0.427326 0.904098i \(-0.359456\pi\)
0.427326 + 0.904098i \(0.359456\pi\)
\(30\) 0 0
\(31\) −324.428 −1.87965 −0.939823 0.341662i \(-0.889010\pi\)
−0.939823 + 0.341662i \(0.889010\pi\)
\(32\) 0 0
\(33\) 199.528i 1.05253i
\(34\) 0 0
\(35\) −205.339 129.408i −0.991673 0.624968i
\(36\) 0 0
\(37\) 73.1277 0.324922 0.162461 0.986715i \(-0.448057\pi\)
0.162461 + 0.986715i \(0.448057\pi\)
\(38\) 0 0
\(39\) 99.2251i 0.407403i
\(40\) 0 0
\(41\) 309.181i 1.17771i −0.808240 0.588854i \(-0.799579\pi\)
0.808240 0.588854i \(-0.200421\pi\)
\(42\) 0 0
\(43\) 28.7269i 0.101879i −0.998702 0.0509396i \(-0.983778\pi\)
0.998702 0.0509396i \(-0.0162216\pi\)
\(44\) 0 0
\(45\) 117.948i 0.390726i
\(46\) 0 0
\(47\) −162.730 −0.505036 −0.252518 0.967592i \(-0.581259\pi\)
−0.252518 + 0.967592i \(0.581259\pi\)
\(48\) 0 0
\(49\) −147.992 309.431i −0.431463 0.902131i
\(50\) 0 0
\(51\) 258.669i 0.710213i
\(52\) 0 0
\(53\) −575.828 −1.49238 −0.746189 0.665734i \(-0.768118\pi\)
−0.746189 + 0.665734i \(0.768118\pi\)
\(54\) 0 0
\(55\) 871.627 2.13691
\(56\) 0 0
\(57\) −63.0188 −0.146439
\(58\) 0 0
\(59\) −861.347 −1.90064 −0.950320 0.311273i \(-0.899245\pi\)
−0.950320 + 0.311273i \(0.899245\pi\)
\(60\) 0 0
\(61\) 367.037i 0.770399i −0.922833 0.385199i \(-0.874133\pi\)
0.922833 0.385199i \(-0.125867\pi\)
\(62\) 0 0
\(63\) 88.8698 141.015i 0.177723 0.282003i
\(64\) 0 0
\(65\) 433.460 0.827140
\(66\) 0 0
\(67\) 818.701i 1.49284i 0.665476 + 0.746419i \(0.268229\pi\)
−0.665476 + 0.746419i \(0.731771\pi\)
\(68\) 0 0
\(69\) 656.006i 1.14455i
\(70\) 0 0
\(71\) 407.234i 0.680701i −0.940299 0.340351i \(-0.889454\pi\)
0.940299 0.340351i \(-0.110546\pi\)
\(72\) 0 0
\(73\) 515.657i 0.826754i −0.910560 0.413377i \(-0.864349\pi\)
0.910560 0.413377i \(-0.135651\pi\)
\(74\) 0 0
\(75\) 140.250 0.215929
\(76\) 0 0
\(77\) 1042.09 + 656.740i 1.54230 + 0.971981i
\(78\) 0 0
\(79\) 1135.55i 1.61721i 0.588355 + 0.808603i \(0.299776\pi\)
−0.588355 + 0.808603i \(0.700224\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −941.158 −1.24464 −0.622322 0.782761i \(-0.713811\pi\)
−0.622322 + 0.782761i \(0.713811\pi\)
\(84\) 0 0
\(85\) 1129.98 1.44193
\(86\) 0 0
\(87\) −400.412 −0.493434
\(88\) 0 0
\(89\) 628.192i 0.748182i 0.927392 + 0.374091i \(0.122045\pi\)
−0.927392 + 0.374091i \(0.877955\pi\)
\(90\) 0 0
\(91\) 518.230 + 326.597i 0.596981 + 0.376227i
\(92\) 0 0
\(93\) 973.285 1.08521
\(94\) 0 0
\(95\) 275.294i 0.297312i
\(96\) 0 0
\(97\) 522.331i 0.546750i −0.961908 0.273375i \(-0.911860\pi\)
0.961908 0.273375i \(-0.0881399\pi\)
\(98\) 0 0
\(99\) 598.583i 0.607676i
\(100\) 0 0
\(101\) 1086.40i 1.07031i 0.844755 + 0.535153i \(0.179746\pi\)
−0.844755 + 0.535153i \(0.820254\pi\)
\(102\) 0 0
\(103\) 196.731 0.188199 0.0940995 0.995563i \(-0.470003\pi\)
0.0940995 + 0.995563i \(0.470003\pi\)
\(104\) 0 0
\(105\) 616.016 + 388.223i 0.572543 + 0.360826i
\(106\) 0 0
\(107\) 870.161i 0.786183i 0.919499 + 0.393092i \(0.128594\pi\)
−0.919499 + 0.393092i \(0.871406\pi\)
\(108\) 0 0
\(109\) −277.220 −0.243604 −0.121802 0.992554i \(-0.538867\pi\)
−0.121802 + 0.992554i \(0.538867\pi\)
\(110\) 0 0
\(111\) −219.383 −0.187594
\(112\) 0 0
\(113\) 958.381 0.797849 0.398924 0.916984i \(-0.369384\pi\)
0.398924 + 0.916984i \(0.369384\pi\)
\(114\) 0 0
\(115\) −2865.73 −2.32375
\(116\) 0 0
\(117\) 297.675i 0.235214i
\(118\) 0 0
\(119\) 1350.97 + 851.402i 1.04070 + 0.655864i
\(120\) 0 0
\(121\) −3092.48 −2.32343
\(122\) 0 0
\(123\) 927.544i 0.679950i
\(124\) 0 0
\(125\) 1025.49i 0.733782i
\(126\) 0 0
\(127\) 63.9927i 0.0447121i −0.999750 0.0223560i \(-0.992883\pi\)
0.999750 0.0223560i \(-0.00711674\pi\)
\(128\) 0 0
\(129\) 86.1807i 0.0588200i
\(130\) 0 0
\(131\) 219.879 0.146648 0.0733240 0.997308i \(-0.476639\pi\)
0.0733240 + 0.997308i \(0.476639\pi\)
\(132\) 0 0
\(133\) 207.425 329.133i 0.135233 0.214582i
\(134\) 0 0
\(135\) 353.844i 0.225586i
\(136\) 0 0
\(137\) −1493.47 −0.931358 −0.465679 0.884954i \(-0.654190\pi\)
−0.465679 + 0.884954i \(0.654190\pi\)
\(138\) 0 0
\(139\) −1149.51 −0.701442 −0.350721 0.936480i \(-0.614064\pi\)
−0.350721 + 0.936480i \(0.614064\pi\)
\(140\) 0 0
\(141\) 488.191 0.291582
\(142\) 0 0
\(143\) −2199.80 −1.28641
\(144\) 0 0
\(145\) 1749.18i 1.00180i
\(146\) 0 0
\(147\) 443.975 + 928.293i 0.249105 + 0.520845i
\(148\) 0 0
\(149\) 229.427 0.126144 0.0630719 0.998009i \(-0.479910\pi\)
0.0630719 + 0.998009i \(0.479910\pi\)
\(150\) 0 0
\(151\) 407.635i 0.219688i 0.993949 + 0.109844i \(0.0350351\pi\)
−0.993949 + 0.109844i \(0.964965\pi\)
\(152\) 0 0
\(153\) 776.007i 0.410042i
\(154\) 0 0
\(155\) 4251.74i 2.20328i
\(156\) 0 0
\(157\) 167.822i 0.0853101i 0.999090 + 0.0426551i \(0.0135817\pi\)
−0.999090 + 0.0426551i \(0.986418\pi\)
\(158\) 0 0
\(159\) 1727.48 0.861625
\(160\) 0 0
\(161\) −3426.17 2159.23i −1.67714 1.05696i
\(162\) 0 0
\(163\) 3096.94i 1.48817i −0.668086 0.744084i \(-0.732886\pi\)
0.668086 0.744084i \(-0.267114\pi\)
\(164\) 0 0
\(165\) −2614.88 −1.23375
\(166\) 0 0
\(167\) −3764.16 −1.74419 −0.872095 0.489337i \(-0.837239\pi\)
−0.872095 + 0.489337i \(0.837239\pi\)
\(168\) 0 0
\(169\) 1103.04 0.502067
\(170\) 0 0
\(171\) 189.056 0.0845468
\(172\) 0 0
\(173\) 3438.40i 1.51108i 0.655103 + 0.755539i \(0.272625\pi\)
−0.655103 + 0.755539i \(0.727375\pi\)
\(174\) 0 0
\(175\) −461.630 + 732.495i −0.199405 + 0.316408i
\(176\) 0 0
\(177\) 2584.04 1.09734
\(178\) 0 0
\(179\) 1128.66i 0.471286i 0.971840 + 0.235643i \(0.0757196\pi\)
−0.971840 + 0.235643i \(0.924280\pi\)
\(180\) 0 0
\(181\) 1613.94i 0.662781i −0.943494 0.331391i \(-0.892482\pi\)
0.943494 0.331391i \(-0.107518\pi\)
\(182\) 0 0
\(183\) 1101.11i 0.444790i
\(184\) 0 0
\(185\) 958.364i 0.380866i
\(186\) 0 0
\(187\) −5734.62 −2.24255
\(188\) 0 0
\(189\) −266.609 + 423.044i −0.102608 + 0.162815i
\(190\) 0 0
\(191\) 2819.97i 1.06830i 0.845388 + 0.534152i \(0.179369\pi\)
−0.845388 + 0.534152i \(0.820631\pi\)
\(192\) 0 0
\(193\) 1749.41 0.652463 0.326232 0.945290i \(-0.394221\pi\)
0.326232 + 0.945290i \(0.394221\pi\)
\(194\) 0 0
\(195\) −1300.38 −0.477549
\(196\) 0 0
\(197\) −3168.95 −1.14608 −0.573041 0.819527i \(-0.694236\pi\)
−0.573041 + 0.819527i \(0.694236\pi\)
\(198\) 0 0
\(199\) −978.805 −0.348672 −0.174336 0.984686i \(-0.555778\pi\)
−0.174336 + 0.984686i \(0.555778\pi\)
\(200\) 0 0
\(201\) 2456.10i 0.861891i
\(202\) 0 0
\(203\) 1317.95 2091.26i 0.455674 0.723043i
\(204\) 0 0
\(205\) −4051.93 −1.38048
\(206\) 0 0
\(207\) 1968.02i 0.660806i
\(208\) 0 0
\(209\) 1397.11i 0.462393i
\(210\) 0 0
\(211\) 1183.70i 0.386204i −0.981179 0.193102i \(-0.938145\pi\)
0.981179 0.193102i \(-0.0618548\pi\)
\(212\) 0 0
\(213\) 1221.70i 0.393003i
\(214\) 0 0
\(215\) −376.476 −0.119421
\(216\) 0 0
\(217\) −3203.54 + 5083.24i −1.00217 + 1.59020i
\(218\) 0 0
\(219\) 1546.97i 0.477327i
\(220\) 0 0
\(221\) −2851.83 −0.868030
\(222\) 0 0
\(223\) −1724.50 −0.517852 −0.258926 0.965897i \(-0.583369\pi\)
−0.258926 + 0.965897i \(0.583369\pi\)
\(224\) 0 0
\(225\) −420.751 −0.124667
\(226\) 0 0
\(227\) 2296.41 0.671445 0.335723 0.941961i \(-0.391020\pi\)
0.335723 + 0.941961i \(0.391020\pi\)
\(228\) 0 0
\(229\) 4404.32i 1.27094i −0.772125 0.635471i \(-0.780806\pi\)
0.772125 0.635471i \(-0.219194\pi\)
\(230\) 0 0
\(231\) −3126.26 1970.22i −0.890446 0.561173i
\(232\) 0 0
\(233\) 4222.07 1.18711 0.593556 0.804793i \(-0.297723\pi\)
0.593556 + 0.804793i \(0.297723\pi\)
\(234\) 0 0
\(235\) 2132.64i 0.591992i
\(236\) 0 0
\(237\) 3406.65i 0.933694i
\(238\) 0 0
\(239\) 4649.91i 1.25848i −0.777210 0.629242i \(-0.783366\pi\)
0.777210 0.629242i \(-0.216634\pi\)
\(240\) 0 0
\(241\) 14.3928i 0.00384697i −0.999998 0.00192348i \(-0.999388\pi\)
0.999998 0.00192348i \(-0.000612264\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −4055.20 + 1939.48i −1.05746 + 0.505751i
\(246\) 0 0
\(247\) 694.783i 0.178980i
\(248\) 0 0
\(249\) 2823.47 0.718596
\(250\) 0 0
\(251\) −6957.71 −1.74967 −0.874834 0.484422i \(-0.839030\pi\)
−0.874834 + 0.484422i \(0.839030\pi\)
\(252\) 0 0
\(253\) 14543.5 3.61400
\(254\) 0 0
\(255\) −3389.94 −0.832496
\(256\) 0 0
\(257\) 4915.09i 1.19298i 0.802622 + 0.596488i \(0.203438\pi\)
−0.802622 + 0.596488i \(0.796562\pi\)
\(258\) 0 0
\(259\) 722.093 1145.79i 0.173238 0.274887i
\(260\) 0 0
\(261\) 1201.24 0.284884
\(262\) 0 0
\(263\) 6930.51i 1.62492i 0.583018 + 0.812459i \(0.301872\pi\)
−0.583018 + 0.812459i \(0.698128\pi\)
\(264\) 0 0
\(265\) 7546.42i 1.74933i
\(266\) 0 0
\(267\) 1884.57i 0.431963i
\(268\) 0 0
\(269\) 481.497i 0.109135i 0.998510 + 0.0545676i \(0.0173780\pi\)
−0.998510 + 0.0545676i \(0.982622\pi\)
\(270\) 0 0
\(271\) −5156.02 −1.15574 −0.577871 0.816128i \(-0.696116\pi\)
−0.577871 + 0.816128i \(0.696116\pi\)
\(272\) 0 0
\(273\) −1554.69 979.790i −0.344667 0.217215i
\(274\) 0 0
\(275\) 3109.31i 0.681814i
\(276\) 0 0
\(277\) 2065.98 0.448134 0.224067 0.974574i \(-0.428067\pi\)
0.224067 + 0.974574i \(0.428067\pi\)
\(278\) 0 0
\(279\) −2919.85 −0.626549
\(280\) 0 0
\(281\) 4260.27 0.904436 0.452218 0.891907i \(-0.350633\pi\)
0.452218 + 0.891907i \(0.350633\pi\)
\(282\) 0 0
\(283\) 3195.17 0.671141 0.335571 0.942015i \(-0.391071\pi\)
0.335571 + 0.942015i \(0.391071\pi\)
\(284\) 0 0
\(285\) 825.883i 0.171653i
\(286\) 0 0
\(287\) −4844.35 3052.98i −0.996351 0.627916i
\(288\) 0 0
\(289\) −2521.40 −0.513209
\(290\) 0 0
\(291\) 1566.99i 0.315666i
\(292\) 0 0
\(293\) 6349.41i 1.26599i 0.774154 + 0.632997i \(0.218176\pi\)
−0.774154 + 0.632997i \(0.781824\pi\)
\(294\) 0 0
\(295\) 11288.2i 2.22789i
\(296\) 0 0
\(297\) 1795.75i 0.350842i
\(298\) 0 0
\(299\) 7232.48 1.39888
\(300\) 0 0
\(301\) −450.102 283.661i −0.0861908 0.0543188i
\(302\) 0 0
\(303\) 3259.20i 0.617942i
\(304\) 0 0
\(305\) −4810.15 −0.903044
\(306\) 0 0
\(307\) 684.156 0.127188 0.0635942 0.997976i \(-0.479744\pi\)
0.0635942 + 0.997976i \(0.479744\pi\)
\(308\) 0 0
\(309\) −590.193 −0.108657
\(310\) 0 0
\(311\) −4151.64 −0.756970 −0.378485 0.925607i \(-0.623555\pi\)
−0.378485 + 0.925607i \(0.623555\pi\)
\(312\) 0 0
\(313\) 2788.88i 0.503632i 0.967775 + 0.251816i \(0.0810277\pi\)
−0.967775 + 0.251816i \(0.918972\pi\)
\(314\) 0 0
\(315\) −1848.05 1164.67i −0.330558 0.208323i
\(316\) 0 0
\(317\) −1123.95 −0.199139 −0.0995697 0.995031i \(-0.531747\pi\)
−0.0995697 + 0.995031i \(0.531747\pi\)
\(318\) 0 0
\(319\) 8877.04i 1.55805i
\(320\) 0 0
\(321\) 2610.48i 0.453903i
\(322\) 0 0
\(323\) 1811.22i 0.312010i
\(324\) 0 0
\(325\) 1546.26i 0.263911i
\(326\) 0 0
\(327\) 831.659 0.140645
\(328\) 0 0
\(329\) −1606.87 + 2549.71i −0.269269 + 0.427265i
\(330\) 0 0
\(331\) 4265.14i 0.708258i −0.935197 0.354129i \(-0.884777\pi\)
0.935197 0.354129i \(-0.115223\pi\)
\(332\) 0 0
\(333\) 658.149 0.108307
\(334\) 0 0
\(335\) 10729.4 1.74987
\(336\) 0 0
\(337\) 5217.89 0.843433 0.421716 0.906728i \(-0.361428\pi\)
0.421716 + 0.906728i \(0.361428\pi\)
\(338\) 0 0
\(339\) −2875.14 −0.460638
\(340\) 0 0
\(341\) 21577.5i 3.42664i
\(342\) 0 0
\(343\) −6309.59 736.671i −0.993253 0.115966i
\(344\) 0 0
\(345\) 8597.19 1.34162
\(346\) 0 0
\(347\) 2596.17i 0.401642i 0.979628 + 0.200821i \(0.0643609\pi\)
−0.979628 + 0.200821i \(0.935639\pi\)
\(348\) 0 0
\(349\) 9467.68i 1.45213i −0.687626 0.726065i \(-0.741347\pi\)
0.687626 0.726065i \(-0.258653\pi\)
\(350\) 0 0
\(351\) 893.026i 0.135801i
\(352\) 0 0
\(353\) 2276.11i 0.343188i 0.985168 + 0.171594i \(0.0548917\pi\)
−0.985168 + 0.171594i \(0.945108\pi\)
\(354\) 0 0
\(355\) −5336.94 −0.797903
\(356\) 0 0
\(357\) −4052.90 2554.20i −0.600847 0.378663i
\(358\) 0 0
\(359\) 4663.75i 0.685636i −0.939402 0.342818i \(-0.888619\pi\)
0.939402 0.342818i \(-0.111381\pi\)
\(360\) 0 0
\(361\) −6417.74 −0.935667
\(362\) 0 0
\(363\) 9277.44 1.34143
\(364\) 0 0
\(365\) −6757.86 −0.969103
\(366\) 0 0
\(367\) 4677.32 0.665270 0.332635 0.943056i \(-0.392062\pi\)
0.332635 + 0.943056i \(0.392062\pi\)
\(368\) 0 0
\(369\) 2782.63i 0.392569i
\(370\) 0 0
\(371\) −5685.97 + 9022.25i −0.795689 + 1.26257i
\(372\) 0 0
\(373\) −551.009 −0.0764884 −0.0382442 0.999268i \(-0.512176\pi\)
−0.0382442 + 0.999268i \(0.512176\pi\)
\(374\) 0 0
\(375\) 3076.48i 0.423649i
\(376\) 0 0
\(377\) 4414.55i 0.603080i
\(378\) 0 0
\(379\) 573.945i 0.0777878i 0.999243 + 0.0388939i \(0.0123834\pi\)
−0.999243 + 0.0388939i \(0.987617\pi\)
\(380\) 0 0
\(381\) 191.978i 0.0258145i
\(382\) 0 0
\(383\) −10376.9 −1.38443 −0.692215 0.721692i \(-0.743365\pi\)
−0.692215 + 0.721692i \(0.743365\pi\)
\(384\) 0 0
\(385\) 8606.81 13656.9i 1.13933 1.80785i
\(386\) 0 0
\(387\) 258.542i 0.0339598i
\(388\) 0 0
\(389\) −8040.94 −1.04805 −0.524025 0.851703i \(-0.675570\pi\)
−0.524025 + 0.851703i \(0.675570\pi\)
\(390\) 0 0
\(391\) 18854.3 2.43862
\(392\) 0 0
\(393\) −659.636 −0.0846673
\(394\) 0 0
\(395\) 14881.8 1.89565
\(396\) 0 0
\(397\) 6608.26i 0.835413i 0.908582 + 0.417706i \(0.137166\pi\)
−0.908582 + 0.417706i \(0.862834\pi\)
\(398\) 0 0
\(399\) −622.274 + 987.398i −0.0780768 + 0.123889i
\(400\) 0 0
\(401\) −292.364 −0.0364089 −0.0182045 0.999834i \(-0.505795\pi\)
−0.0182045 + 0.999834i \(0.505795\pi\)
\(402\) 0 0
\(403\) 10730.5i 1.32636i
\(404\) 0 0
\(405\) 1061.53i 0.130242i
\(406\) 0 0
\(407\) 4863.67i 0.592342i
\(408\) 0 0
\(409\) 1334.80i 0.161374i −0.996740 0.0806868i \(-0.974289\pi\)
0.996740 0.0806868i \(-0.0257114\pi\)
\(410\) 0 0
\(411\) 4480.42 0.537720
\(412\) 0 0
\(413\) −8505.30 + 13495.8i −1.01336 + 1.60796i
\(414\) 0 0
\(415\) 12334.2i 1.45894i
\(416\) 0 0
\(417\) 3448.54 0.404978
\(418\) 0 0
\(419\) −8852.36 −1.03214 −0.516069 0.856547i \(-0.672605\pi\)
−0.516069 + 0.856547i \(0.672605\pi\)
\(420\) 0 0
\(421\) 6270.01 0.725847 0.362924 0.931819i \(-0.381779\pi\)
0.362924 + 0.931819i \(0.381779\pi\)
\(422\) 0 0
\(423\) −1464.57 −0.168345
\(424\) 0 0
\(425\) 4030.93i 0.460068i
\(426\) 0 0
\(427\) −5750.86 3624.28i −0.651764 0.410752i
\(428\) 0 0
\(429\) 6599.39 0.742707
\(430\) 0 0
\(431\) 205.022i 0.0229131i 0.999934 + 0.0114566i \(0.00364682\pi\)
−0.999934 + 0.0114566i \(0.996353\pi\)
\(432\) 0 0
\(433\) 12003.1i 1.33217i 0.745875 + 0.666086i \(0.232032\pi\)
−0.745875 + 0.666086i \(0.767968\pi\)
\(434\) 0 0
\(435\) 5247.54i 0.578392i
\(436\) 0 0
\(437\) 4593.41i 0.502821i
\(438\) 0 0
\(439\) 2757.16 0.299754 0.149877 0.988705i \(-0.452112\pi\)
0.149877 + 0.988705i \(0.452112\pi\)
\(440\) 0 0
\(441\) −1331.93 2784.88i −0.143821 0.300710i
\(442\) 0 0
\(443\) 3626.94i 0.388987i 0.980904 + 0.194493i \(0.0623063\pi\)
−0.980904 + 0.194493i \(0.937694\pi\)
\(444\) 0 0
\(445\) 8232.67 0.877002
\(446\) 0 0
\(447\) −688.282 −0.0728291
\(448\) 0 0
\(449\) −16303.5 −1.71360 −0.856801 0.515647i \(-0.827552\pi\)
−0.856801 + 0.515647i \(0.827552\pi\)
\(450\) 0 0
\(451\) 20563.4 2.14699
\(452\) 0 0
\(453\) 1222.90i 0.126837i
\(454\) 0 0
\(455\) 4280.16 6791.58i 0.441005 0.699768i
\(456\) 0 0
\(457\) 11048.3 1.13089 0.565447 0.824784i \(-0.308704\pi\)
0.565447 + 0.824784i \(0.308704\pi\)
\(458\) 0 0
\(459\) 2328.02i 0.236738i
\(460\) 0 0
\(461\) 5823.95i 0.588391i 0.955745 + 0.294196i \(0.0950518\pi\)
−0.955745 + 0.294196i \(0.904948\pi\)
\(462\) 0 0
\(463\) 15638.6i 1.56974i −0.619661 0.784869i \(-0.712730\pi\)
0.619661 0.784869i \(-0.287270\pi\)
\(464\) 0 0
\(465\) 12755.2i 1.27206i
\(466\) 0 0
\(467\) 730.574 0.0723917 0.0361959 0.999345i \(-0.488476\pi\)
0.0361959 + 0.999345i \(0.488476\pi\)
\(468\) 0 0
\(469\) 12827.7 + 8084.19i 1.26296 + 0.795935i
\(470\) 0 0
\(471\) 503.467i 0.0492538i
\(472\) 0 0
\(473\) 1910.60 0.185729
\(474\) 0 0
\(475\) −982.045 −0.0948617
\(476\) 0 0
\(477\) −5182.45 −0.497459
\(478\) 0 0
\(479\) 1211.82 0.115594 0.0577971 0.998328i \(-0.481592\pi\)
0.0577971 + 0.998328i \(0.481592\pi\)
\(480\) 0 0
\(481\) 2418.70i 0.229279i
\(482\) 0 0
\(483\) 10278.5 + 6477.68i 0.968299 + 0.610238i
\(484\) 0 0
\(485\) −6845.33 −0.640888
\(486\) 0 0
\(487\) 5068.72i 0.471634i −0.971798 0.235817i \(-0.924223\pi\)
0.971798 0.235817i \(-0.0757766\pi\)
\(488\) 0 0
\(489\) 9290.83i 0.859194i
\(490\) 0 0
\(491\) 11619.0i 1.06794i −0.845503 0.533971i \(-0.820699\pi\)
0.845503 0.533971i \(-0.179301\pi\)
\(492\) 0 0
\(493\) 11508.2i 1.05133i
\(494\) 0 0
\(495\) 7844.64 0.712304
\(496\) 0 0
\(497\) −6380.67 4021.20i −0.575879 0.362928i
\(498\) 0 0
\(499\) 6132.84i 0.550187i 0.961417 + 0.275094i \(0.0887089\pi\)
−0.961417 + 0.275094i \(0.911291\pi\)
\(500\) 0 0
\(501\) 11292.5 1.00701
\(502\) 0 0
\(503\) 5868.53 0.520208 0.260104 0.965581i \(-0.416243\pi\)
0.260104 + 0.965581i \(0.416243\pi\)
\(504\) 0 0
\(505\) 14237.7 1.25459
\(506\) 0 0
\(507\) −3309.13 −0.289869
\(508\) 0 0
\(509\) 8716.70i 0.759059i 0.925180 + 0.379529i \(0.123914\pi\)
−0.925180 + 0.379529i \(0.876086\pi\)
\(510\) 0 0
\(511\) −8079.47 5091.81i −0.699442 0.440799i
\(512\) 0 0
\(513\) −567.169 −0.0488131
\(514\) 0 0
\(515\) 2578.23i 0.220603i
\(516\) 0 0
\(517\) 10823.1i 0.920693i
\(518\) 0 0
\(519\) 10315.2i 0.872422i
\(520\) 0 0
\(521\) 6289.58i 0.528890i −0.964401 0.264445i \(-0.914811\pi\)
0.964401 0.264445i \(-0.0851887\pi\)
\(522\) 0 0
\(523\) −19253.1 −1.60971 −0.804854 0.593473i \(-0.797757\pi\)
−0.804854 + 0.593473i \(0.797757\pi\)
\(524\) 0 0
\(525\) 1384.89 2197.48i 0.115127 0.182678i
\(526\) 0 0
\(527\) 27973.2i 2.31220i
\(528\) 0 0
\(529\) −35649.1 −2.92998
\(530\) 0 0
\(531\) −7752.12 −0.633547
\(532\) 0 0
\(533\) 10226.2 0.831041
\(534\) 0 0
\(535\) 11403.8 0.921546
\(536\) 0 0
\(537\) 3385.99i 0.272097i
\(538\) 0 0
\(539\) 20580.0 9842.82i 1.64461 0.786568i
\(540\) 0 0
\(541\) 6806.51 0.540915 0.270457 0.962732i \(-0.412825\pi\)
0.270457 + 0.962732i \(0.412825\pi\)
\(542\) 0 0
\(543\) 4841.83i 0.382657i
\(544\) 0 0
\(545\) 3633.06i 0.285547i
\(546\) 0 0
\(547\) 2333.60i 0.182409i −0.995832 0.0912044i \(-0.970928\pi\)
0.995832 0.0912044i \(-0.0290717\pi\)
\(548\) 0 0
\(549\) 3303.34i 0.256800i
\(550\) 0 0
\(551\) 2803.72 0.216774
\(552\) 0 0
\(553\) 17792.1 + 11212.9i 1.36817 + 0.862243i
\(554\) 0 0
\(555\) 2875.09i 0.219893i
\(556\) 0 0
\(557\) −3035.04 −0.230878 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(558\) 0 0
\(559\) 950.143 0.0718904
\(560\) 0 0
\(561\) 17203.9 1.29474
\(562\) 0 0
\(563\) −9471.73 −0.709034 −0.354517 0.935050i \(-0.615355\pi\)
−0.354517 + 0.935050i \(0.615355\pi\)
\(564\) 0 0
\(565\) 12559.9i 0.935220i
\(566\) 0 0
\(567\) 799.828 1269.13i 0.0592409 0.0940010i
\(568\) 0 0
\(569\) −4580.84 −0.337502 −0.168751 0.985659i \(-0.553973\pi\)
−0.168751 + 0.985659i \(0.553973\pi\)
\(570\) 0 0
\(571\) 15298.8i 1.12125i 0.828068 + 0.560627i \(0.189440\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(572\) 0 0
\(573\) 8459.92i 0.616786i
\(574\) 0 0
\(575\) 10222.8i 0.741426i
\(576\) 0 0
\(577\) 22630.9i 1.63282i −0.577473 0.816410i \(-0.695961\pi\)
0.577473 0.816410i \(-0.304039\pi\)
\(578\) 0 0
\(579\) −5248.23 −0.376700
\(580\) 0 0
\(581\) −9293.38 + 14746.3i −0.663605 + 1.05298i
\(582\) 0 0
\(583\) 38297.9i 2.72065i
\(584\) 0 0
\(585\) 3901.14 0.275713
\(586\) 0 0
\(587\) −17070.7 −1.20031 −0.600157 0.799883i \(-0.704895\pi\)
−0.600157 + 0.799883i \(0.704895\pi\)
\(588\) 0 0
\(589\) −6815.02 −0.476754
\(590\) 0 0
\(591\) 9506.84 0.661691
\(592\) 0 0
\(593\) 2588.94i 0.179283i 0.995974 + 0.0896416i \(0.0285722\pi\)
−0.995974 + 0.0896416i \(0.971428\pi\)
\(594\) 0 0
\(595\) 11157.9 17704.9i 0.768790 1.21988i
\(596\) 0 0
\(597\) 2936.42 0.201306
\(598\) 0 0
\(599\) 432.117i 0.0294755i 0.999891 + 0.0147378i \(0.00469134\pi\)
−0.999891 + 0.0147378i \(0.995309\pi\)
\(600\) 0 0
\(601\) 11299.7i 0.766927i 0.923556 + 0.383463i \(0.125269\pi\)
−0.923556 + 0.383463i \(0.874731\pi\)
\(602\) 0 0
\(603\) 7368.31i 0.497613i
\(604\) 0 0
\(605\) 40528.0i 2.72347i
\(606\) 0 0
\(607\) 20570.8 1.37552 0.687761 0.725938i \(-0.258594\pi\)
0.687761 + 0.725938i \(0.258594\pi\)
\(608\) 0 0
\(609\) −3953.84 + 6273.79i −0.263083 + 0.417449i
\(610\) 0 0
\(611\) 5382.32i 0.356375i
\(612\) 0 0
\(613\) −11492.6 −0.757231 −0.378615 0.925554i \(-0.623600\pi\)
−0.378615 + 0.925554i \(0.623600\pi\)
\(614\) 0 0
\(615\) 12155.8 0.797022
\(616\) 0 0
\(617\) −26688.7 −1.74140 −0.870701 0.491812i \(-0.836335\pi\)
−0.870701 + 0.491812i \(0.836335\pi\)
\(618\) 0 0
\(619\) 4597.10 0.298502 0.149251 0.988799i \(-0.452314\pi\)
0.149251 + 0.988799i \(0.452314\pi\)
\(620\) 0 0
\(621\) 5904.06i 0.381516i
\(622\) 0 0
\(623\) 9842.70 + 6203.03i 0.632969 + 0.398907i
\(624\) 0 0
\(625\) −19283.2 −1.23412
\(626\) 0 0
\(627\) 4191.33i 0.266963i
\(628\) 0 0
\(629\) 6305.28i 0.399695i
\(630\) 0 0
\(631\) 5107.87i 0.322252i 0.986934 + 0.161126i \(0.0515126\pi\)
−0.986934 + 0.161126i \(0.948487\pi\)
\(632\) 0 0
\(633\) 3551.09i 0.222975i
\(634\) 0 0
\(635\) −838.647 −0.0524105
\(636\) 0 0
\(637\) 10234.4 4894.83i 0.636583 0.304459i
\(638\) 0 0
\(639\) 3665.11i 0.226900i
\(640\) 0 0
\(641\) 1581.52 0.0974512 0.0487256 0.998812i \(-0.484484\pi\)
0.0487256 + 0.998812i \(0.484484\pi\)
\(642\) 0 0
\(643\) −2210.56 −0.135577 −0.0677884 0.997700i \(-0.521594\pi\)
−0.0677884 + 0.997700i \(0.521594\pi\)
\(644\) 0 0
\(645\) 1129.43 0.0689475
\(646\) 0 0
\(647\) −12507.1 −0.759976 −0.379988 0.924991i \(-0.624072\pi\)
−0.379988 + 0.924991i \(0.624072\pi\)
\(648\) 0 0
\(649\) 57287.5i 3.46492i
\(650\) 0 0
\(651\) 9610.62 15249.7i 0.578602 0.918101i
\(652\) 0 0
\(653\) −990.222 −0.0593421 −0.0296710 0.999560i \(-0.509446\pi\)
−0.0296710 + 0.999560i \(0.509446\pi\)
\(654\) 0 0
\(655\) 2881.59i 0.171898i
\(656\) 0 0
\(657\) 4640.91i 0.275585i
\(658\) 0 0
\(659\) 25900.2i 1.53100i −0.643435 0.765501i \(-0.722491\pi\)
0.643435 0.765501i \(-0.277509\pi\)
\(660\) 0 0
\(661\) 22104.7i 1.30071i −0.759628 0.650357i \(-0.774619\pi\)
0.759628 0.650357i \(-0.225381\pi\)
\(662\) 0 0
\(663\) 8555.48 0.501157
\(664\) 0 0
\(665\) −4313.40 2718.37i −0.251528 0.158517i
\(666\) 0 0
\(667\) 29185.9i 1.69428i
\(668\) 0 0
\(669\) 5173.49 0.298982
\(670\) 0 0
\(671\) 24411.4 1.40446
\(672\) 0 0
\(673\) −6847.62 −0.392209 −0.196104 0.980583i \(-0.562829\pi\)
−0.196104 + 0.980583i \(0.562829\pi\)
\(674\) 0 0
\(675\) 1262.25 0.0719765
\(676\) 0 0
\(677\) 8884.62i 0.504378i −0.967678 0.252189i \(-0.918850\pi\)
0.967678 0.252189i \(-0.0811504\pi\)
\(678\) 0 0
\(679\) −8184.05 5157.72i −0.462555 0.291510i
\(680\) 0 0
\(681\) −6889.23 −0.387659
\(682\) 0 0
\(683\) 24173.3i 1.35427i 0.735859 + 0.677135i \(0.236779\pi\)
−0.735859 + 0.677135i \(0.763221\pi\)
\(684\) 0 0
\(685\) 19572.5i 1.09172i
\(686\) 0 0
\(687\) 13213.0i 0.733778i
\(688\) 0 0
\(689\) 19045.5i 1.05309i
\(690\) 0 0
\(691\) 11158.2 0.614296 0.307148 0.951662i \(-0.400625\pi\)
0.307148 + 0.951662i \(0.400625\pi\)
\(692\) 0 0
\(693\) 9378.79 + 5910.66i 0.514099 + 0.323994i
\(694\) 0 0
\(695\) 15064.8i 0.822215i
\(696\) 0 0
\(697\) 26658.5 1.44873
\(698\) 0 0
\(699\) −12666.2 −0.685380
\(700\) 0 0
\(701\) −4440.80 −0.239268 −0.119634 0.992818i \(-0.538172\pi\)
−0.119634 + 0.992818i \(0.538172\pi\)
\(702\) 0 0
\(703\) 1536.14 0.0824134
\(704\) 0 0
\(705\) 6397.92i 0.341786i
\(706\) 0 0
\(707\) 17022.1 + 10727.6i 0.905489 + 0.570654i
\(708\) 0 0
\(709\) −19194.5 −1.01673 −0.508367 0.861140i \(-0.669751\pi\)
−0.508367 + 0.861140i \(0.669751\pi\)
\(710\) 0 0
\(711\) 10219.9i 0.539068i
\(712\) 0 0
\(713\) 70942.3i 3.72624i
\(714\) 0 0
\(715\) 28829.1i 1.50790i
\(716\) 0 0
\(717\) 13949.7i 0.726586i
\(718\) 0 0
\(719\) 35198.1 1.82569 0.912843 0.408310i \(-0.133882\pi\)
0.912843 + 0.408310i \(0.133882\pi\)
\(720\) 0 0
\(721\) 1942.61 3082.44i 0.100342 0.159218i
\(722\) 0 0
\(723\) 43.1783i 0.00222105i
\(724\) 0 0
\(725\) −6239.77 −0.319641
\(726\) 0 0
\(727\) 13232.3 0.675047 0.337524 0.941317i \(-0.390411\pi\)
0.337524 + 0.941317i \(0.390411\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 2476.92 0.125324
\(732\) 0 0
\(733\) 20075.6i 1.01161i −0.862649 0.505804i \(-0.831196\pi\)
0.862649 0.505804i \(-0.168804\pi\)
\(734\) 0 0
\(735\) 12165.6 5818.45i 0.610523 0.291995i
\(736\) 0 0
\(737\) −54451.2 −2.72149
\(738\) 0 0
\(739\) 15808.7i 0.786917i 0.919342 + 0.393459i \(0.128722\pi\)
−0.919342 + 0.393459i \(0.871278\pi\)
\(740\) 0 0
\(741\) 2084.35i 0.103334i
\(742\) 0 0
\(743\) 4785.70i 0.236299i −0.992996 0.118150i \(-0.962304\pi\)
0.992996 0.118150i \(-0.0376962\pi\)
\(744\) 0 0
\(745\) 3006.73i 0.147863i
\(746\) 0 0
\(747\) −8470.42 −0.414881
\(748\) 0 0
\(749\) 13633.9 + 8592.33i 0.665118 + 0.419168i
\(750\) 0 0
\(751\) 26285.5i 1.27719i 0.769543 + 0.638595i \(0.220484\pi\)
−0.769543 + 0.638595i \(0.779516\pi\)
\(752\) 0 0
\(753\) 20873.1 1.01017
\(754\) 0 0
\(755\) 5342.20 0.257513
\(756\) 0 0
\(757\) −26989.1 −1.29582 −0.647910 0.761717i \(-0.724357\pi\)
−0.647910 + 0.761717i \(0.724357\pi\)
\(758\) 0 0
\(759\) −43630.5 −2.08654
\(760\) 0 0
\(761\) 26835.7i 1.27831i 0.769079 + 0.639154i \(0.220715\pi\)
−0.769079 + 0.639154i \(0.779285\pi\)
\(762\) 0 0
\(763\) −2737.38 + 4343.56i −0.129882 + 0.206091i
\(764\) 0 0
\(765\) 10169.8 0.480642
\(766\) 0 0
\(767\) 28489.1i 1.34117i
\(768\) 0 0
\(769\) 7482.83i 0.350894i −0.984489 0.175447i \(-0.943863\pi\)
0.984489 0.175447i \(-0.0561371\pi\)
\(770\) 0 0
\(771\) 14745.3i 0.688765i
\(772\) 0 0
\(773\) 755.076i 0.0351335i −0.999846 0.0175668i \(-0.994408\pi\)
0.999846 0.0175668i \(-0.00559196\pi\)
\(774\) 0 0
\(775\) 15167.1 0.702989
\(776\) 0 0
\(777\) −2166.28 + 3437.36i −0.100019 + 0.158706i
\(778\) 0 0
\(779\) 6494.74i 0.298714i
\(780\) 0 0
\(781\) 27084.8 1.24094
\(782\) 0 0
\(783\) −3603.71 −0.164478
\(784\) 0 0
\(785\) 2199.37 0.0999987
\(786\) 0 0
\(787\) 38108.4 1.72607 0.863036 0.505143i \(-0.168560\pi\)
0.863036 + 0.505143i \(0.168560\pi\)
\(788\) 0 0
\(789\) 20791.5i 0.938147i
\(790\) 0 0
\(791\) 9463.45 15016.2i 0.425388 0.674987i
\(792\) 0 0
\(793\) 12139.8 0.543627
\(794\) 0 0
\(795\) 22639.3i 1.00998i
\(796\) 0 0
\(797\) 20013.0i 0.889455i −0.895666 0.444728i \(-0.853301\pi\)
0.895666 0.444728i \(-0.146699\pi\)
\(798\) 0 0
\(799\) 14031.1i 0.621257i
\(800\) 0 0
\(801\) 5653.72i 0.249394i
\(802\) 0 0
\(803\) 34295.9 1.50720
\(804\) 0 0
\(805\) −28297.4 + 44901.1i −1.23895 + 1.96591i
\(806\) 0 0
\(807\) 1444.49i 0.0630093i
\(808\) 0 0
\(809\) 15288.5 0.664420 0.332210 0.943206i \(-0.392206\pi\)
0.332210 + 0.943206i \(0.392206\pi\)
\(810\) 0 0
\(811\) −15348.1 −0.664542 −0.332271 0.943184i \(-0.607815\pi\)
−0.332271 + 0.943184i \(0.607815\pi\)
\(812\) 0 0
\(813\) 15468.1 0.667268
\(814\) 0 0
\(815\) −40586.5 −1.74440
\(816\) 0 0
\(817\) 603.444i 0.0258407i
\(818\) 0 0
\(819\) 4664.07 + 2939.37i 0.198994 + 0.125409i
\(820\) 0 0
\(821\) 15311.7 0.650891 0.325445 0.945561i \(-0.394486\pi\)
0.325445 + 0.945561i \(0.394486\pi\)
\(822\) 0 0
\(823\) 30450.5i 1.28972i −0.764301 0.644859i \(-0.776916\pi\)
0.764301 0.644859i \(-0.223084\pi\)
\(824\) 0 0
\(825\) 9327.94i 0.393645i
\(826\) 0 0
\(827\) 8352.35i 0.351197i 0.984462 + 0.175598i \(0.0561860\pi\)
−0.984462 + 0.175598i \(0.943814\pi\)
\(828\) 0 0
\(829\) 18715.9i 0.784114i 0.919941 + 0.392057i \(0.128236\pi\)
−0.919941 + 0.392057i \(0.871764\pi\)
\(830\) 0 0
\(831\) −6197.95 −0.258730
\(832\) 0 0
\(833\) 26680.0 12760.3i 1.10973 0.530753i
\(834\) 0 0
\(835\) 49330.6i 2.04450i
\(836\) 0 0
\(837\) 8759.56 0.361738
\(838\) 0 0
\(839\) 19309.6 0.794566 0.397283 0.917696i \(-0.369953\pi\)
0.397283 + 0.917696i \(0.369953\pi\)
\(840\) 0 0
\(841\) −6574.54 −0.269570
\(842\) 0 0
\(843\) −12780.8 −0.522177
\(844\) 0 0
\(845\) 14455.7i 0.588512i
\(846\) 0 0
\(847\) −30536.5 + 48453.9i −1.23878 + 1.96564i
\(848\) 0 0
\(849\) −9585.50 −0.387484
\(850\) 0 0
\(851\) 15990.7i 0.644131i
\(852\) 0 0
\(853\) 27138.9i 1.08935i −0.838646 0.544677i \(-0.816652\pi\)
0.838646 0.544677i \(-0.183348\pi\)
\(854\) 0 0
\(855\) 2477.65i 0.0991039i
\(856\) 0 0
\(857\) 25760.4i 1.02679i 0.858152 + 0.513395i \(0.171613\pi\)
−0.858152 + 0.513395i \(0.828387\pi\)
\(858\) 0 0
\(859\) 42892.5 1.70369 0.851846 0.523792i \(-0.175483\pi\)
0.851846 + 0.523792i \(0.175483\pi\)
\(860\) 0 0
\(861\) 14533.0 + 9158.95i 0.575243 + 0.362528i
\(862\) 0 0
\(863\) 26017.9i 1.02626i 0.858312 + 0.513128i \(0.171513\pi\)
−0.858312 + 0.513128i \(0.828487\pi\)
\(864\) 0 0
\(865\) 45061.4 1.77125
\(866\) 0 0
\(867\) 7564.19 0.296301
\(868\) 0 0
\(869\) −75524.5 −2.94821
\(870\) 0 0
\(871\) −27078.6 −1.05341
\(872\) 0 0
\(873\) 4700.98i 0.182250i
\(874\) 0 0
\(875\) −16067.7 10126.1i −0.620786 0.391230i
\(876\) 0 0
\(877\) 4450.82 0.171372 0.0856861 0.996322i \(-0.472692\pi\)
0.0856861 + 0.996322i \(0.472692\pi\)
\(878\) 0 0
\(879\) 19048.2i 0.730922i
\(880\) 0 0
\(881\) 15737.7i 0.601837i −0.953650 0.300918i \(-0.902707\pi\)
0.953650 0.300918i \(-0.0972931\pi\)
\(882\) 0 0
\(883\) 5677.78i 0.216390i −0.994130 0.108195i \(-0.965493\pi\)
0.994130 0.108195i \(-0.0345071\pi\)
\(884\) 0 0
\(885\) 33864.7i 1.28627i
\(886\) 0 0
\(887\) 7460.00 0.282393 0.141196 0.989982i \(-0.454905\pi\)
0.141196 + 0.989982i \(0.454905\pi\)
\(888\) 0 0
\(889\) −1002.66 631.891i −0.0378268 0.0238391i
\(890\) 0 0
\(891\) 5387.25i 0.202559i
\(892\) 0 0
\(893\) −3418.36 −0.128097
\(894\) 0 0
\(895\) 14791.5 0.552431
\(896\) 0 0
\(897\) −21697.4 −0.807644
\(898\) 0 0
\(899\) −43301.7 −1.60644
\(900\) 0 0
\(901\) 49649.6i 1.83581i
\(902\) 0 0
\(903\) 1350.30 + 850.984i 0.0497623 + 0.0313610i
\(904\) 0 0
\(905\) −21151.3 −0.776897
\(906\) 0 0
\(907\) 34731.8i 1.27150i 0.771894 + 0.635751i \(0.219309\pi\)
−0.771894 + 0.635751i \(0.780691\pi\)
\(908\) 0 0
\(909\) 9777.61i 0.356769i
\(910\) 0 0
\(911\) 1692.78i 0.0615636i −0.999526 0.0307818i \(-0.990200\pi\)
0.999526 0.0307818i \(-0.00979970\pi\)
\(912\) 0 0
\(913\) 62595.7i 2.26902i
\(914\) 0 0
\(915\) 14430.5 0.521373
\(916\) 0 0
\(917\) 2171.17 3445.13i 0.0781881 0.124066i
\(918\) 0 0
\(919\) 7532.50i 0.270375i −0.990820 0.135187i \(-0.956836\pi\)
0.990820 0.135187i \(-0.0431636\pi\)
\(920\) 0 0
\(921\) −2052.47 −0.0734323
\(922\) 0 0
\(923\) 13469.3 0.480332
\(924\) 0 0
\(925\) −3418.73 −0.121521
\(926\) 0 0
\(927\) 1770.58 0.0627330
\(928\) 0 0
\(929\) 21633.4i 0.764012i −0.924160 0.382006i \(-0.875233\pi\)
0.924160 0.382006i \(-0.124767\pi\)
\(930\) 0 0
\(931\) −3108.75 6499.99i −0.109436 0.228817i
\(932\) 0 0
\(933\) 12454.9 0.437037
\(934\) 0 0
\(935\) 75154.2i 2.62867i
\(936\) 0 0
\(937\) 33386.1i 1.16401i 0.813185 + 0.582005i \(0.197732\pi\)
−0.813185 + 0.582005i \(0.802268\pi\)
\(938\) 0 0
\(939\) 8366.63i 0.290772i
\(940\) 0 0
\(941\) 22939.9i 0.794709i −0.917665 0.397354i \(-0.869928\pi\)
0.917665 0.397354i \(-0.130072\pi\)
\(942\) 0 0
\(943\) −67608.3 −2.33471
\(944\) 0 0
\(945\) 5544.14 + 3494.01i 0.190848 + 0.120275i
\(946\) 0 0
\(947\) 3074.78i 0.105509i −0.998608 0.0527545i \(-0.983200\pi\)
0.998608 0.0527545i \(-0.0168001\pi\)
\(948\) 0 0
\(949\) 17055.4 0.583394
\(950\) 0 0
\(951\) 3371.84 0.114973
\(952\) 0 0
\(953\) 26530.3 0.901786 0.450893 0.892578i \(-0.351106\pi\)
0.450893 + 0.892578i \(0.351106\pi\)
\(954\) 0 0
\(955\) 36956.7 1.25224
\(956\) 0 0
\(957\) 26631.1i 0.899543i
\(958\) 0 0
\(959\) −14747.2 + 23400.2i −0.496571 + 0.787937i
\(960\) 0 0
\(961\) 75462.6 2.53307
\(962\) 0 0
\(963\) 7831.45i 0.262061i
\(964\) 0 0
\(965\) 22926.6i 0.764803i
\(966\) 0 0
\(967\) 5995.46i 0.199381i 0.995019 + 0.0996903i \(0.0317852\pi\)
−0.995019 + 0.0996903i \(0.968215\pi\)
\(968\) 0 0
\(969\) 5433.67i 0.180139i
\(970\) 0 0
\(971\) −16996.3 −0.561726 −0.280863 0.959748i \(-0.590621\pi\)
−0.280863 + 0.959748i \(0.590621\pi\)
\(972\) 0 0
\(973\) −11350.8 + 18010.9i −0.373987 + 0.593427i
\(974\) 0 0
\(975\) 4638.78i 0.152369i
\(976\) 0 0
\(977\) 8989.54 0.294371 0.147186 0.989109i \(-0.452979\pi\)
0.147186 + 0.989109i \(0.452979\pi\)
\(978\) 0 0
\(979\) −41780.6 −1.36396
\(980\) 0 0
\(981\) −2494.98 −0.0812013
\(982\) 0 0
\(983\) 1169.17 0.0379356 0.0189678 0.999820i \(-0.493962\pi\)
0.0189678 + 0.999820i \(0.493962\pi\)
\(984\) 0 0
\(985\) 41530.1i 1.34341i
\(986\) 0 0
\(987\) 4820.61 7649.13i 0.155463 0.246681i
\(988\) 0 0
\(989\) −6281.67 −0.201967
\(990\) 0 0
\(991\) 55448.1i 1.77736i −0.458525 0.888681i \(-0.651622\pi\)
0.458525 0.888681i \(-0.348378\pi\)
\(992\) 0 0
\(993\) 12795.4i 0.408913i
\(994\) 0 0
\(995\) 12827.6i 0.408705i
\(996\) 0 0
\(997\) 48292.0i 1.53403i 0.641631 + 0.767013i \(0.278258\pi\)
−0.641631 + 0.767013i \(0.721742\pi\)
\(998\) 0 0
\(999\) −1974.45 −0.0625313
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 1344.4.b.i.895.5 24
4.3 odd 2 1344.4.b.j.895.5 24
7.6 odd 2 1344.4.b.j.895.20 24
8.3 odd 2 672.4.b.a.223.20 yes 24
8.5 even 2 672.4.b.b.223.20 yes 24
28.27 even 2 inner 1344.4.b.i.895.20 24
56.13 odd 2 672.4.b.a.223.5 24
56.27 even 2 672.4.b.b.223.5 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.4.b.a.223.5 24 56.13 odd 2
672.4.b.a.223.20 yes 24 8.3 odd 2
672.4.b.b.223.5 yes 24 56.27 even 2
672.4.b.b.223.20 yes 24 8.5 even 2
1344.4.b.i.895.5 24 1.1 even 1 trivial
1344.4.b.i.895.20 24 28.27 even 2 inner
1344.4.b.j.895.5 24 4.3 odd 2
1344.4.b.j.895.20 24 7.6 odd 2